#72240913

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting

#72217511 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#72215837
commutator, cosocle	C₂×C₂²⋊(C₂²⋊A₄)	C₂⁴
Frattini, socle	C₂³	C₂³×#1020
Fitting	#72217511	D₃
hypercenter	C₂²	#72215837
hypocenter	C₂²⋊(C₂²⋊A₄)	C₂²×D₄
LCS	G ▷ C₂×C₂²⋊(C₂²⋊A₄) ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂ ◁ C₂²

Only the 176 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3,2^6|c^b=ce,f^d=f^2,g^b=gk,g^d=gh,g^f=h,h^b=hl,h^f=gh,i^a=ikl,i^b=ik,i^d=gj,i^f=j,j^a=jk,j^b=jl,j^d=ghi,j^f=ij,k^d=kl,k^f=l,l^f=kl

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

automorphism group of

C₂×B(4,𝔽₂), C₂×#458

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 11, 4, F.11*F.12);;
gap> SetConjugate(coll, 11, 6, F.12);;
gap> SetConjugate(coll, 12, 6, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);