D₂₃₃₁₂ #72280342

order	46624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₃₁₂

Splittings

Fitting	C₁₆×C₃₁×C₄₇ ⋊ C₂
Hall	C₃₁×C₄₇ ⋊ D₁₆
Sylow	C₄₇ ⋊ D₄₉₆
Sylow	C₃₁ ⋊ D₇₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₁₆₅₆
commutator, cosocle	C₈×C₃₁×C₄₇	C₂²
Frattini	C₈	C₂×D₁₄₅₇
Fitting	C₁₆×C₃₁×C₄₇	C₂
socle	C₂×C₃₁×C₄₇	D₈
hypercenter	C₁₆	D₁₄₅₇
hypocenter	C₃₁×C₄₇	D₁₆
LCS	G ▷ C₈×C₃₁×C₄₇ ▷ C₄×C₃₁×C₄₇ ▷ C₂×C₃₁×C₄₇ ▷ C₃₁×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 94
transitive degree	μ_t(G) = 23312

A minimal faithful permutation representation — 2 generators on 94 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^5,31,47|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^30,g^a=g^46

Conjugacy classes & character table

11659 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, 2, 31, 47]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 6, 1, F.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D23312 #72280342