Finite Group Databasebeta

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C3×#14136 #10632743

order768
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C42

Aliases

direct productC3×#14136

Subgroups

subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

2742 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)(9,11)(12,13,15,14)(16,17,20,18)(19,22,23,21);(5,7)(9,11)(17,18)(21,22);(4,5)(6,7)(8,9,10,11)(12,14,15,13)(16,18,20,17)(19,22,23,21);(16,19,20,23)(17,21,18,22)

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270856, #72270885, D3×#14136, …
maximal subgroup of#72270856, #72270885, C3×#647667, …
maximal normal subgroup of#72270856, #72270885, C3×#647667, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7124);