D₁₁₁₃₂ #72215823

order	22264
automorphism	Aut(G) order 53878880
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₃₂

Splittings

Fitting	C₄×C₂₃×C₁₂₁ ⋊ C₂
Hall	C₂₃×C₁₂₁ ⋊ D₄
Sylow	C₁₂₁ ⋊ D₉₂
Sylow	C₂₃ ⋊ D₄₈₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₇₈₃
commutator, cosocle	C₂×C₂₃×C₁₂₁	C₂²
Frattini	C₂×C₁₁	C₂×D₂₅₃
Fitting	C₄×C₂₃×C₁₂₁	C₂
socle	C₂×C₁₁×C₂₃	C₂×D₁₁
hypercenter	C₄	D₂₇₈₃
hypocenter	C₂₃×C₁₂₁	D₄
LCS	G ▷ C₂×C₂₃×C₁₂₁ ▷ C₂₃×C₁₂₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 148
transitive degree	μ_t(G) = 11132

A minimal faithful permutation representation — 2 generators on 148 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);(2,4)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)(134,141)(135,140)(136,139)(137,138)

Polycyclic presentation:

2^3,11,23,11|b^2=c,d^11=f,b^a=bc,d^a=d^10f^10,e^a=e^22,f^a=f^10

Conjugacy classes & character table

5569 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₂₆₄
maximal normal subgroup of	D₂₂₂₆₄
maximal quotient of	Dic₁₁₁₃₂, D₂₂₂₆₄

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 11, 23, 11]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10*F.6^10);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11132 #72215823