Finite Group Databasebeta

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C3×#10710712 #72252315

order2304
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC3×#10710712

Splittings

FittingC32×#1781 ⋊ C2
SylowC32 ⋊ #12705

Subgroups

subgroupG/H
centerC2×C3#69498
commutatorC24×C3C22×C3×C4
FrattiniC22×D4C22×C3×D3
FittingC32×#1781C2
socleC2×C32#1781
prime radicalC3×#87239C2
cosocleC22×C3×D4C23×C3
hypercenterC3×#1781D3
hypocenterC3C3×#12705
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C23×C3×D4 ◁ C3×#1781

2812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252320, #72252338, C3×#72231563
maximal normal subgroup of#72252320, #72252338, C3×#72231563

GAP

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

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