#72341908

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

Aliases

No aliases for this group.

Splittings

Fitting

C₂²×#442 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×S₄
commutator, cosocle	C₂×C₄×A₄	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	C₂²×#442	D₃
socle	C₂⁴	C₂×D₃×D₄
hypercenter	#442	S₄
hypocenter	A₄	#3109
LCS	G ▷ C₂×C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

2857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 20 points:

(1,4,3,2)(5,8,9,6)(7,10,11,12)(13,15,17,16)(14,19,20,18);(2,3,4)(5,6,9,8)(7,10,11,12)(13,15,17,16)(14,19,20,18);(5,7,9,11)(6,10,8,12)(14,18)(15,16)(19,20);(5,6,9,8)(7,10,11,12)(13,14,17,20)(15,19,16,18)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72342092

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