D₁₂₁₂₄ #72224038

order	24248
automorphism	Aut(G) order 62850816
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₁₂₄

Splittings

Fitting	C₄×C₇×C₄₃₃ ⋊ C₂
Hall	C₇×C₄₃₃ ⋊ D₄
Sylow	C₄₃₃ ⋊ D₂₈
Sylow	C₇ ⋊ D₁₇₃₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₀₃₁
commutator, socle, cosocle	C₂×C₇×C₄₃₃	C₂²
Fitting	C₄×C₇×C₄₃₃	C₂
hypercenter	C₄	D₃₀₃₁
hypocenter	C₇×C₄₃₃	D₄
LCS	G ▷ C₂×C₇×C₄₃₃ ▷ C₇×C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 444
transitive degree	μ_t(G) = 12124

A minimal faithful permutation representation — 2 generators on 444 points:

(1,2,3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444);(2,4)(6,11)(7,10)(8,9)(13,444)(14,443)(15,442)(16,441)(17,440)(18,439)(19,438)(20,437)(21,436)(22,435)(23,434)(24,433)(25,432)(26,431)(27,430)(28,429)(29,428)(30,427)(31,426)(32,425)(33,424)(34,423)(35,422)(36,421)(37,420)(38,419)(39,418)(40,417)(41,416)(42,415)(43,414)(44,413)(45,412)(46,411)(47,410)(48,409)(49,408)(50,407)(51,406)(52,405)(53,404)(54,403)(55,402)(56,401)(57,400)(58,399)(59,398)(60,397)(61,396)(62,395)(63,394)(64,393)(65,392)(66,391)(67,390)(68,389)(69,388)(70,387)(71,386)(72,385)(73,384)(74,383)(75,382)(76,381)(77,380)(78,379)(79,378)(80,377)(81,376)(82,375)(83,374)(84,373)(85,372)(86,371)(87,370)(88,369)(89,368)(90,367)(91,366)(92,365)(93,364)(94,363)(95,362)(96,361)(97,360)(98,359)(99,358)(100,357)(101,356)(102,355)(103,354)(104,353)(105,352)(106,351)(107,350)(108,349)(109,348)(110,347)(111,346)(112,345)(113,344)(114,343)(115,342)(116,341)(117,340)(118,339)(119,338)(120,337)(121,336)(122,335)(123,334)(124,333)(125,332)(126,331)(127,330)(128,329)(129,328)(130,327)(131,326)(132,325)(133,324)(134,323)(135,322)(136,321)(137,320)(138,319)(139,318)(140,317)(141,316)(142,315)(143,314)(144,313)(145,312)(146,311)(147,310)(148,309)(149,308)(150,307)(151,306)(152,305)(153,304)(154,303)(155,302)(156,301)(157,300)(158,299)(159,298)(160,297)(161,296)(162,295)(163,294)(164,293)(165,292)(166,291)(167,290)(168,289)(169,288)(170,287)(171,286)(172,285)(173,284)(174,283)(175,282)(176,281)(177,280)(178,279)(179,278)(180,277)(181,276)(182,275)(183,274)(184,273)(185,272)(186,271)(187,270)(188,269)(189,268)(190,267)(191,266)(192,265)(193,264)(194,263)(195,262)(196,261)(197,260)(198,259)(199,258)(200,257)(201,256)(202,255)(203,254)(204,253)(205,252)(206,251)(207,250)(208,249)(209,248)(210,247)(211,246)(212,245)(213,244)(214,243)(215,242)(216,241)(217,240)(218,239)(219,238)(220,237)(221,236)(222,235)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229)

Polycyclic presentation:

2^3,7,433|b^2=c,b^a=bc,d^a=d^6,e^a=e^432

Conjugacy classes & character table

6065 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₄₂₄₈
maximal normal subgroup of	D₂₄₂₄₈
maximal quotient of	Dic₁₂₁₂₄, D₂₄₂₄₈

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 433]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^432);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D12124 #72224038