#72258604

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#418 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×S₄
commutator, cosocle	C₂×C₄×A₄	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	C₂²×#418	D₃
socle	C₂⁴	C₂×D₃×D₄
hypercenter	#418	S₄
hypocenter	A₄	#3038
LCS	G ▷ C₂×C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

3757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 16 points:

(1,3,4,2)(5,6,7,8);(10,14)(13,16);(6,8)(10,13)(11,15)(14,16);(3,4)(6,8)(9,10,11,13,12,14,15,16)

Polycyclic presentation:

2^7,3,2^2|a^2=e,d^2=f,e^2=g,b^a=beg,c^a=cg,d^a=df,d^b=df,e^b=eg,h^a=h^2,h^d=h^2,i^h=j,j^a=ij,j^d=ij,j^h=ij

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258631
maximal normal subgroup of	#72258631

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