D₁₁₆₅₆ #72223932

order	23312
automorphism	Aut(G) order 64341120
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₆₅₆

Splittings

Fitting	C₈×C₃₁×C₄₇ ⋊ C₂
Hall	C₃₁×C₄₇ ⋊ D₈
Sylow	C₄₇ ⋊ D₂₄₈
Sylow	C₃₁ ⋊ D₃₇₆

Subgroups

	subgroup	G/H
center	C₂	D₅₈₂₈
commutator, cosocle	C₄×C₃₁×C₄₇	C₂²
Frattini	C₄	C₂×D₁₄₅₇
Fitting	C₈×C₃₁×C₄₇	C₂
socle	C₂×C₃₁×C₄₇	D₄
hypercenter	C₈	D₁₄₅₇
hypocenter	C₃₁×C₄₇	D₈
LCS	G ▷ C₄×C₃₁×C₄₇ ▷ C₂×C₃₁×C₄₇ ▷ C₃₁×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 11656

A minimal faithful permutation representation — 2 generators on 86 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);(2,8)(3,7)(4,6)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^4,31,47|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^30,f^a=f^46

Conjugacy classes & character table

5831 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₃₁₂
maximal normal subgroup of	D₂₃₃₁₂
maximal quotient of	Dic₁₁₆₅₆, D₂₃₃₁₂

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, 31, 47]);;
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.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11656 #72223932