Finite Group Databasebeta

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C3×#23573 #10642180

order768
nilpotency class3
isoclinismisoclinic class of #23391
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#23573

Subgroups

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

1508 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)(8,9)(10,13)(11,12)(14,15)(16,17)(18,23)(19,21)(20,22);(4,5)(6,7)(9,12)(13,15)(17,21)(22,23);(5,7)(17,22)(18,20)(21,23);(8,10,11,14)(9,13,12,15)(16,18,19,20)(17,23,21,22)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#23573, #72269976
maximal subgroup ofD3×#23573, #72269976, C3×#6453969
maximal normal subgroup ofD3×#23573, #72269976, C3×#6453969
maximal quotient ofC3×#499269

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16561);