#72240986

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#6372744 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#11713272
commutator, cosocle	#5950	C₂⁴
Frattini	C₂²×C₄	C₂³×S₄
Fitting	#6372744	D₃
hypercenter	M₁₆	#1020⋊C₂
hypocenter	#997	C₂²×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

6189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,27,26,8,17,10,11,32,18,2,31,30,6,25,13,20,15,19,3,23,5,7,28,22)(4,16,12,29,14,9,24,21);(2,7)(8,20)(9,16)(10,19)(21,29)(22,30)(23,32)(25,27);(2,8)(3,11)(4,15)(7,20)(9,23)(10,22)(12,28)(13,26)(14,17)(16,32)(19,30)(21,27)(24,31)(25,29);(2,7)(4,14)(5,18)(8,20)(12,24)(13,26)(15,17)(28,31)

Polycyclic presentation:

2^5,3,2^5|d^2=e,e^2=k,g^2=k,h^2=k,d^a=dk,d^b=de,d^c=dk,e^b=ek,f^b=f^2,g^a=gijk,g^b=ghjk,g^d=gijk,g^f=h,h^a=hik,h^b=hi,h^d=hik,h^f=ghk,h^g=hk,i^f=j,i^g=ik,j^b=ij,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245616, C₂×#72240986
maximal normal subgroup of	#72245616, C₂×#72240986
maximal quotient of	C₂×#72240986

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", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);