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C4×#64269⋊C3 #72269092

order3456
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24×C3

Aliases

direct productC4×#64269⋊C3

Splittings

FittingC24×C32×C4 ⋊ D3

Subgroups

subgroupG/H
centerC2×C4C2×(C3⋊S4)⋊C3
commutatorC2×C3×A4C22×C3×C4
FrattiniC22×C3C22×C3×S4
FittingC24×C32×C4D3
socleC24×C3C22×C3×D3
prime radicalC2×#64269⋊C3C2
cosocleC22×C3×A4C23×C3
hypercenterC22×C4(C3⋊S4)⋊C3
hypocenterC3×A4C3×C4×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C4 ◁ C22×C4

2211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,4)(10,12)(13,14,20)(15,17,19)(16,18,21);(2,3)(9,10)(11,12)(14,16)(15,18)(17,19)(20,21);(5,6,7,8)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#72213301

GAP

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

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