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C3×#37660 #10656267

order768
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#37660

Subgroups

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

4078 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)(8,9)(10,14)(11,12)(13,15)(16,17)(18,22)(19,21)(20,23);(5,7)(9,12)(13,14)(16,17,19,21)(18,20,23,22);(4,5)(6,7)(16,18)(17,20)(19,23)(21,22);(8,10)(9,13)(11,15)(12,14)(16,18)(17,20)(19,23)(21,22);(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,20)(19,21)(22,23)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#37660, #72248532, #72253627
maximal subgroup ofD3×#37660, C3×#7792761, #72248532, …
maximal normal subgroup ofD3×#37660, C3×#7792761, #72248532, …
maximal quotient ofC3×#846041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 30648);