D₁₁₁₇₆ #72215853

order	22352
automorphism	Aut(G) order 56327040
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₇₆

Splittings

Fitting	C₈×C₁₁×C₁₂₇ ⋊ C₂
Hall	C₁₁×C₁₂₇ ⋊ D₈
Sylow	C₁₂₇ ⋊ D₈₈
Sylow	C₁₁ ⋊ D₁₀₁₆

Subgroups

	subgroup	G/H
center	C₂	D₅₅₈₈
commutator, cosocle	C₄×C₁₁×C₁₂₇	C₂²
Frattini	C₄	C₂×D₁₃₉₇
Fitting	C₈×C₁₁×C₁₂₇	C₂
socle	C₂×C₁₁×C₁₂₇	D₄
hypercenter	C₈	D₁₃₉₇
hypocenter	C₁₁×C₁₂₇	D₈
LCS	G ▷ C₄×C₁₁×C₁₂₇ ▷ C₂×C₁₁×C₁₂₇ ▷ C₁₁×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 146
transitive degree	μ_t(G) = 11176

A minimal faithful permutation representation — 2 generators on 146 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);(2,8)(3,7)(4,6)(10,19)(11,18)(12,17)(13,16)(14,15)(21,146)(22,145)(23,144)(24,143)(25,142)(26,141)(27,140)(28,139)(29,138)(30,137)(31,136)(32,135)(33,134)(34,133)(35,132)(36,131)(37,130)(38,129)(39,128)(40,127)(41,126)(42,125)(43,124)(44,123)(45,122)(46,121)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)

Polycyclic presentation:

2^4,11,127|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^10,f^a=f^126

Conjugacy classes & character table

5591 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₃₅₂
maximal normal subgroup of	D₂₂₃₅₂
maximal quotient of	Dic₁₁₁₇₆, D₂₂₃₅₂

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, 2, 11, 127]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11176 #72215853