D₁₁₉₂₄ #72223993

order	23848
automorphism	Aut(G) order 64389600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₉₂₄

Splittings

Fitting	C₄×C₁₁×C₂₇₁ ⋊ C₂
Hall	C₁₁×C₂₇₁ ⋊ D₄
Sylow	C₂₇₁ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₁₀₈₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₈₁
commutator, socle, cosocle	C₂×C₁₁×C₂₇₁	C₂²
Fitting	C₄×C₁₁×C₂₇₁	C₂
hypercenter	C₄	D₂₉₈₁
hypocenter	C₁₁×C₂₇₁	D₄
LCS	G ▷ C₂×C₁₁×C₂₇₁ ▷ C₁₁×C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 286
transitive degree	μ_t(G) = 11924

A minimal faithful permutation representation — 2 generators on 286 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);(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,286)(18,285)(19,284)(20,283)(21,282)(22,281)(23,280)(24,279)(25,278)(26,277)(27,276)(28,275)(29,274)(30,273)(31,272)(32,271)(33,270)(34,269)(35,268)(36,267)(37,266)(38,265)(39,264)(40,263)(41,262)(42,261)(43,260)(44,259)(45,258)(46,257)(47,256)(48,255)(49,254)(50,253)(51,252)(52,251)(53,250)(54,249)(55,248)(56,247)(57,246)(58,245)(59,244)(60,243)(61,242)(62,241)(63,240)(64,239)(65,238)(66,237)(67,236)(68,235)(69,234)(70,233)(71,232)(72,231)(73,230)(74,229)(75,228)(76,227)(77,226)(78,225)(79,224)(80,223)(81,222)(82,221)(83,220)(84,219)(85,218)(86,217)(87,216)(88,215)(89,214)(90,213)(91,212)(92,211)(93,210)(94,209)(95,208)(96,207)(97,206)(98,205)(99,204)(100,203)(101,202)(102,201)(103,200)(104,199)(105,198)(106,197)(107,196)(108,195)(109,194)(110,193)(111,192)(112,191)(113,190)(114,189)(115,188)(116,187)(117,186)(118,185)(119,184)(120,183)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,176)(128,175)(129,174)(130,173)(131,172)(132,171)(133,170)(134,169)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(141,162)(142,161)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)

Polycyclic presentation:

2^3,11,271|b^2=c,b^a=bc,d^a=d^10,e^a=e^270

Conjugacy classes & character table

5965 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₈₄₈
maximal normal subgroup of	D₂₃₈₄₈
maximal quotient of	Dic₁₁₉₂₄, D₂₃₈₄₈

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, 11, 271]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^270);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11924 #72223993