#72259155

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁸ ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×S₄
commutator, cosocle	C₂³×A₄	C₂⁴
Fitting	C₂⁸	D₃
socle	C₂⁵	C₂³×D₃
hypercenter	C₂⁶	S₄
hypocenter	A₄	#2846
LCS	G ▷ C₂³×A₄ ▷ A₄
UCS	C₁ ◁ C₂³ ◁ C₂⁶

79748 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 79748 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^7,3,2^2|b^a=bf,c^b=ce,d^b=dg,h^b=h^2,i^b=j,i^h=j,j^b=i,j^h=ij

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#120×GL(2,ℤ/4)
maximal normal subgroup of	#120×GL(2,ℤ/4)

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