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C3×#24196 #10642803

order768
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#24196

Subgroups

subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

1428 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)(4,5)(6,7)(8,9)(10,14)(11,12)(13,15)(17,21)(18,19)(22,23);(9,12)(13,14)(17,22)(18,19)(21,23);(5,7)(16,17)(18,22)(19,21)(20,23);(8,10,11,15)(9,13,12,14)(17,23)(21,22)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72269965, D3×#24196
maximal subgroup of#72269965, D3×#24196, C3×#6453969
maximal normal subgroup of#72269965, D3×#24196, C3×#6453969
maximal quotient ofC3×#500650

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17184);