diff --git a/src/core/group/group.h b/src/core/group/group.h
index 2b2fe74..8e076b4 100644
--- a/src/core/group/group.h
+++ b/src/core/group/group.h
@@ -394,8 +394,10 @@ private:
 
 #include "internal/type_id.inl"
 #include "internal/group_union.inl"
-#include "internal/group_cases.inl"
+
 #include "internal/group.inl"
-#include "internal/case_info.inl"
 #include "internal/group_mirror.inl"
+
+#include "internal/group_cases.inl"
+#include "internal/case_info.inl"
 #include "internal/hash.inl"
diff --git a/src/core/group/internal/case_info.inl b/src/core/group/internal/case_info.inl
index aa2102e..baf668f 100644
--- a/src/core/group/internal/case_info.inl
+++ b/src/core/group/internal/case_info.inl
@@ -9,6 +9,10 @@ inline std::ostream& operator<<(std::ostream &out, CaseInfo self) {
     return out;
 }
 
+constexpr auto operator==(const CaseInfo &lhs, const CaseInfo &rhs) {
+    return lhs.group_ == rhs.group_ && lhs.case_id_ == rhs.case_id_;
+}
+
 inline std::string CaseInfo::to_string() const {
     return std::format("{}-{}", group_.to_string(), case_id_);
 }
@@ -21,17 +25,17 @@ constexpr uint32_t CaseInfo::case_id() const {
     return case_id_;
 }
 
-constexpr std::optional<CaseInfo> CaseInfo::create(Group group, uint32_t case_id) {
+constexpr std::optional<CaseInfo> CaseInfo::create(const Group group, const uint32_t case_id) {
     if (case_id >= group.size()) {
         return std::nullopt;
     }
     return unsafe_create(group, case_id);
 }
 
-constexpr CaseInfo CaseInfo::unsafe_create(Group group, uint32_t case_id) {
+constexpr CaseInfo CaseInfo::unsafe_create(const Group group, const uint32_t case_id) {
     return {group, case_id};
 }
 
-constexpr CaseInfo::CaseInfo(Group group, uint32_t case_id) : group_(group), case_id_(case_id) {}
+constexpr CaseInfo::CaseInfo(const Group group, const uint32_t case_id) : group_(group), case_id_(case_id) {}
 
 } // namespace klotski::group
diff --git a/src/core/group/internal/group.inl b/src/core/group/internal/group.inl
index a1f3171..a7e0d9f 100644
--- a/src/core/group/internal/group.inl
+++ b/src/core/group/internal/group.inl
@@ -26,20 +26,22 @@ constexpr char Group::toward_char() const {
         case MirrorType::Horizontal:
             return (toward_ == Toward::A) ? 'n' : 'u';
         case MirrorType::Centro:
-            return (toward_ == Toward::A) ? '?' : '!'; // TODO: select chars for centro
+            return (toward_ == Toward::A) ? 'h' : 's';
         case MirrorType::Vertical:
             return (toward_ == Toward::A) ? 'p' : 'q';
         case MirrorType::Ordinary:
             if (toward_ == Toward::A) {
                 return 'a';
-            } else if (toward_ == Toward::B) {
+            }
+            if (toward_ == Toward::B) {
                 return 'b';
-            } else if (toward_ == Toward::C) {
+            }
+            if (toward_ == Toward::C) {
                 return 'c';
-            } else if (toward_ == Toward::D) {
+            }
+            if (toward_ == Toward::D) {
                 return 'd';
             }
-            return '\0'; // TODO: never reach
     }
     return '\0'; // TODO: never reach
 }
@@ -47,7 +49,7 @@ constexpr char Group::toward_char() const {
 // ----------------------------------------------------------------------------------------- //
 
 #ifndef KLSK_NDEBUG
-inline std::ostream& operator<<(std::ostream &out, Group self) {
+inline std::ostream& operator<<(std::ostream &out, const Group self) {
     out << self.to_string();
     return out;
 }
@@ -62,16 +64,16 @@ inline std::string Group::to_string() const {
 }
 
 constexpr auto operator==(const Group &lhs, const Group &rhs) {
-    return lhs.type_id_ == rhs.type_id_
-        && lhs.pattern_id_ == rhs.pattern_id_
-        && lhs.toward_ == rhs.toward_;
+    return lhs.toward_ == rhs.toward_
+        && lhs.type_id_ == rhs.type_id_
+        && lhs.pattern_id_ == rhs.pattern_id_;
 }
 
-constexpr Group Group::unsafe_create(uint_fast8_t type_id, uint_fast16_t pattern_id, Toward toward) {
+constexpr Group Group::unsafe_create(const uint_fast8_t type_id, const uint_fast16_t pattern_id, const Toward toward) {
     return {toward, type_id, pattern_id};
 }
 
-constexpr std::optional<Group> Group::create(uint_fast8_t type_id, uint_fast16_t pattern_id, Toward toward) {
+constexpr std::optional<Group> Group::create(const uint_fast8_t type_id, const uint_fast16_t pattern_id, const Toward toward) {
     if (type_id >= TYPE_ID_LIMIT) {
         return std::nullopt;
     }
@@ -90,11 +92,11 @@ constexpr uint32_t Group::size() const {
     return (PATTERN_DATA[flat_id()] >> 3) & 0xFFFFF;
 }
 
-inline Group Group::from_short_code(codec::ShortCode short_code) {
+inline Group Group::from_short_code(const codec::ShortCode short_code) {
     return from_common_code(short_code.to_common_code());
 }
 
-inline Group Group::from_common_code(codec::CommonCode common_code) {
+inline Group Group::from_common_code(const codec::CommonCode common_code) {
     return from_raw_code(common_code.to_raw_code());
 }
 
diff --git a/src/core/group/internal/group_cases.cc b/src/core/group/internal/group_cases.cc
index 89d03b2..6499986 100644
--- a/src/core/group/internal/group_cases.cc
+++ b/src/core/group/internal/group_cases.cc
@@ -131,7 +131,7 @@ CommonCode GroupCases::fast_obtain_code(CaseInfo info) {
     return CommonCode::unsafe_create(head << 32 | range);
 }
 
-CaseInfo GroupCases::fast_obtain_info(ShortCode short_code) {
+CaseInfo GroupCases::fast_obtain_info(const ShortCode short_code) {
     uint16_t type_id = GroupUnion::from_short_code(short_code).unwrap(); // NOTE: need to convert as CommonCode
     uint16_t pattern_id = (*rev_data)[short_code.unwrap()].pattern_id;
     uint16_t toward_id = (*rev_data)[short_code.unwrap()].toward_id;
@@ -141,7 +141,7 @@ CaseInfo GroupCases::fast_obtain_info(ShortCode short_code) {
     return CaseInfo::unsafe_create(group, case_id);
 }
 
-CaseInfo GroupCases::fast_obtain_info(CommonCode common_code) {
+CaseInfo GroupCases::fast_obtain_info(const CommonCode common_code) {
     auto short_code = common_code.to_short_code();
     uint16_t type_id = GroupUnion::from_common_code(common_code).unwrap();
     uint16_t pattern_id = (*rev_data)[short_code.unwrap()].pattern_id;
@@ -152,14 +152,14 @@ CaseInfo GroupCases::fast_obtain_info(CommonCode common_code) {
     return CaseInfo::unsafe_create(group, case_id);
 }
 
-Group GroupCases::fast_obtain_group(codec::ShortCode short_code) {
+Group GroupCases::fast_obtain_group(const ShortCode short_code) {
     uint16_t type_id = GroupUnion::from_short_code(short_code).unwrap();
     uint16_t pattern_id = (*rev_data)[short_code.unwrap()].pattern_id;
     uint16_t toward_id = (*rev_data)[short_code.unwrap()].toward_id;
     return Group::unsafe_create(type_id, pattern_id, (Group::Toward)toward_id);
 }
 
-Group GroupCases::fast_obtain_group(codec::CommonCode common_code) {
+Group GroupCases::fast_obtain_group(const CommonCode common_code) {
     auto short_code = common_code.to_short_code();
     uint16_t type_id = GroupUnion::from_common_code(common_code).unwrap();
     uint16_t pattern_id = (*rev_data)[short_code.unwrap()].pattern_id;
@@ -227,7 +227,7 @@ static std::vector<RawCode> Group_extend_for_obtain_info(RawCode raw_code) {
     return codes;
 }
 
-CaseInfo GroupCases::tiny_obtain_info(CommonCode common_code) {
+CaseInfo GroupCases::tiny_obtain_info(const CommonCode common_code) {
     auto raw_codes = Group_extend_for_obtain_info(common_code.to_raw_code());
     std::vector<CommonCode> common_codes;
     common_codes.reserve(raw_codes.size());
diff --git a/src/core/group/internal/group_cases.inl b/src/core/group/internal/group_cases.inl
index a6b1535..c243d8a 100644
--- a/src/core/group/internal/group_cases.inl
+++ b/src/core/group/internal/group_cases.inl
@@ -9,28 +9,28 @@ inline codec::CommonCode GroupCases::obtain_code(CaseInfo info) {
     return tiny_obtain_code(info);
 }
 
-inline CaseInfo GroupCases::obtain_info(codec::CommonCode common_code) {
+inline CaseInfo GroupCases::obtain_info(const codec::CommonCode common_code) {
     if (fast_) {
         return fast_obtain_info(common_code);
     }
     return tiny_obtain_info(common_code);
 }
 
-inline Group GroupCases::obtain_group(codec::CommonCode common_code) {
+inline Group GroupCases::obtain_group(const codec::CommonCode common_code) {
     if (fast_) {
         return fast_obtain_group(common_code);
     }
     return Group::from_common_code(common_code);
 }
 
-inline Group GroupCases::obtain_group(codec::ShortCode short_code) {
+inline Group GroupCases::obtain_group(const codec::ShortCode short_code) {
     if (fast_) {
         return fast_obtain_group(short_code);
     }
     return Group::from_short_code(short_code);
 }
 
-inline CaseInfo GroupCases::obtain_info(codec::ShortCode short_code) {
+inline CaseInfo GroupCases::obtain_info(const codec::ShortCode short_code) {
     if (fast_) {
         return fast_obtain_info(short_code);
     }