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C3×#36714 #10655321

order768
nilpotency class2
isoclinismisoclinic class of #36702
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4

Aliases

direct productC3×#36714

Subgroups

subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

4170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#36714, #72270673
maximal subgroup ofD3×#36714, #72270673, C3×#7659307
maximal normal subgroup ofD3×#36714, #72270673, C3×#7659307
maximal quotient ofC3×#616960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 29702);