Finite Group Databasebeta

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C3×#10826736 #72252328

order2304
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#10826736

Splittings

FittingC32×#3484 ⋊ C2
SylowC32 ⋊ #23895

Subgroups

subgroupG/H
centerC2×C3#84052
commutator, cosocleC24×C3C24×C3
FrattiniC24C23×C3×D3
FittingC32×#3484C2
socleC2×C32#2403
hypercenterC3×#3484D3
hypocenterC3C3×#23895
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C24×C3 ◁ C3×#3484

4954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252331, #72252356, C3×#72231563
maximal normal subgroup of#72252331, #72252356, 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)(7,9,11,16)(8,12,14,20)(10,17,18,15)(13,21,22,19), (7,10)(8,13,14,22)(9,15)(11,18)(12,19,20,21)(16,17), (7,9)(8,12)(10,15)(11,16)(13,21)(14,20)(17,18)(19,22), (5,6)(7,8)(9,12)(10,13,18,22)(11,14)(15,21,17,19)(16,20));