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C3×#10984 #10629591

order768
nilpotency class3
isoclinismisoclinic class of #8325
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC3×#10984

Subgroups

subgroupG/H
center, socleC22×C3#379
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC2×C3×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1462 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)(8,9)(10,17,19,14)(11,13)(12,16)(15,20)(18,22,23,21);(4,5)(6,7)(9,13)(11,20)(14,21)(15,16)(17,22)(18,23);(5,7)(8,10)(9,14)(11,18)(12,19)(13,21)(15,22)(16,17)(20,23)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248636, D3×#10984, #72248900, …
maximal subgroup of#72248636, D3×#10984, C3×#417597, …
maximal normal subgroup of#72248636, D3×#10984, C3×#417597, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3972);