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C3×#23913 #10642520

order768
nilpotency class3
isoclinismisoclinic class of #23912
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#23913

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72271572
maximal subgroup ofC3×#6437449, #72271572
maximal normal subgroup ofC3×#6437449, #72271572

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16901);