Finite Group Databasebeta

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A4×#12067 #72342724

order3072
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24

Aliases

direct productA4×#12067

Splittings

Fitting, SylowC22×#12067 ⋊ C3

Subgroups

subgroupG/H
centerC22A4×#442
commutatorC22×C8C22×C3×C8
FrattiniC4×C8C23×A4
FittingC22×#12067C3
socleC24C3×#442
prime radicalC2×A4×D16C4
cosocleC22×C4×C8C23×C3
hypercenter#12067A4
hypocenterC22C3×#12067
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C23 ▷ C22
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #12067

1585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 3 generators on 28 points:

(2,3,4)(5,7,12,11,9,10,8,6)(13,15,25,24,18,17,16,26)(14,19,23,21,22,27,20,28);(1,4)(2,3)(14,19)(15,24)(16,25)(17,26)(20,21)(22,27)(23,28);(5,6,8,10,9,11,12,7)(13,14,16,20,18,22,25,23)(15,19,26,28,17,27,24,21)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×#12067, #72342940, A4×AGL(1,ℤ/32)
maximal normal subgroup ofS4×#12067, #72342940, A4×AGL(1,ℤ/32)
maximal quotient ofA4×AGL(1,ℤ/32)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,4)(5,7,12,11,9,10,8,6)(13,15,25,24,18,17,16,26)(14,19,23,21,22,27,20,28), (1,4)(2,3)(14,19)(15,24)(16,25)(17,26)(20,21)(22,27)(23,28), (5,6,8,10,9,11,12,7)(13,14,16,20,18,22,25,23)(15,19,26,28,17,27,24,21));