Finite Group Databasebeta

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C3×#10953 #10629560

order768
nilpotency class3
isoclinismisoclinic class of #1902
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC3×#10953

Subgroups

subgroupG/H
center, socleC22×C3#386
commutatorC2×C4C2×C3×C42
Frattini, cosocleC23×C4C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248313, #72248337, #72248382, …
maximal subgroup of#72248313, #72248337, C3×#421068, …
maximal normal subgroup of#72248313, #72248337, C3×#421068, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3941);