D₃₁₃₂₈ #72301023

order	62656
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₃₂₈

Splittings

Fitting	C₁₁×C₃₂×C₈₉ ⋊ C₂
Hall	C₁₁×C₈₉ ⋊ D₃₂
Sylow	C₈₉ ⋊ D₃₅₂
Sylow	C₁₁ ⋊ D₂₈₄₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₆₆₄
commutator, cosocle	C₁₁×C₁₆×C₈₉	C₂²
Frattini	C₁₆	C₂×D₉₇₉
Fitting	C₁₁×C₃₂×C₈₉	C₂
socle	C₂×C₁₁×C₈₉	D₁₆
hypercenter	C₃₂	D₉₇₉
hypocenter	C₁₁×C₈₉	D₃₂
LCS	G ▷ C₁₁×C₁₆×C₈₉ ▷ C₈×C₁₁×C₈₉ ▷ C₄×C₁₁×C₈₉ ▷ C₂×C₁₁×C₈₉ ▷ C₁₁×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 31328

A minimal faithful permutation representation — 2 generators on 132 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,43)(35,42)(36,41)(37,40)(38,39)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)

Polycyclic presentation:

2^6,11,89|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^10,h^a=h^88

Conjugacy classes & character table

15667 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 11, 89]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31328 #72301023