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C3×#21379 #10639986

order768
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC3×#21379

Subgroups

subgroupG/H
centerC22×C3×C4C24
commutatorC23C23×C3×C4
Frattini, cosocleC22×C4C24×C3
socleC23×C3C23×C4
prime radicalC22×C3×#41C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#21379, #72270035
maximal subgroup ofC3×#6453969, D3×#21379, #72270035
maximal normal subgroup ofC3×#6453969, D3×#21379, #72270035
maximal quotient ofC3×#495505

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 14367);