D₂₈₇₇₆ #72292398

order	57552
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₇₇₆

Splittings

Fitting	C₃×C₈×C₁₁×C₁₀₉ ⋊ C₂
Hall	C₃×C₁₁×C₁₀₉ ⋊ D₈
Hall	C₁₁×C₁₀₉ ⋊ D₂₄
Hall	C₃×C₁₀₉ ⋊ D₈₈
Sylow	C₁₀₉ ⋊ D₂₆₄
Hall	C₃×C₁₁ ⋊ D₈₇₂
Sylow	C₁₁ ⋊ D₂₆₁₆
Sylow	C₃ ⋊ D₉₅₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₄₃₈₈
commutator, cosocle	C₃×C₄×C₁₁×C₁₀₉	C₂²
Frattini	C₄	C₂×D₃₅₉₇
Fitting	C₃×C₈×C₁₁×C₁₀₉	C₂
socle	C₂×C₃×C₁₁×C₁₀₉	D₄
hypercenter	C₈	D₃₅₉₇
hypocenter	C₃×C₁₁×C₁₀₉	D₈
LCS	G ▷ C₃×C₄×C₁₁×C₁₀₉ ▷ C₂×C₃×C₁₁×C₁₀₉ ▷ C₃×C₁₁×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 28776

A minimal faithful permutation representation — 2 generators on 131 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);(2,8)(3,7)(4,6)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2^4,3,11,109|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^10,g^a=g^108

Conjugacy classes & character table

14391 conjugacy classes. The character table is too large to display.

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 11, 109]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^108);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28776 #72292398