#72211558

order	1536
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂⁵

solvable

Aliases

No aliases for this group.

Splittings

derived

C₂²⋊(C₂²⋊SL(2,𝔽₃)) ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	#87588
commutator, hypocenter	C₂²⋊(C₂²⋊SL(2,𝔽₃))	C₄
Fitting	C₂⁴×#41	D₃
socle	C₂⁵	SL(2,ℤ/4)
prime radical, cosocle	#11710913	C₂
LCS	G ▷ C₂²⋊(C₂²⋊SL(2,𝔽₃))
UCS	C₁ ◁ C₄

6618 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 6618 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^7|a^2=b,b^2=j,h^2=j,i^2=j,c^a=c^2,d^a=de,d^c=e,e^c=de,f^a=g,f^c=g,g^a=f,g^c=fg,h^a=fhi,h^c=i,i^a=fgij,i^c=hij,i^h=ij

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211581, #72212876, C₂×#72211558, …
maximal normal subgroup of	#72211581, #72212876, C₂×#72211558, …
maximal quotient of	C₂×#72211558

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