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C3×#20739 #10639346

order768
nilpotency class3
isoclinismisoclinic class of #3003
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC3×#20739

Subgroups

subgroupG/H
center, socleC23×C3C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

1738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

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

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72269891, D3×#20739
maximal subgroup of#72269891, D3×#20739, C3×#6453969
maximal normal subgroup of#72269891, D3×#20739, C3×#6453969
maximal quotient ofC3×#494058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13727);