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C3×#40606 #10659213

order768
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#40606

Subgroups

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

2630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

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

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#40606, #72253575
maximal subgroup ofD3×#40606, #72253575, C3×#7918389
maximal normal subgroup ofD3×#40606, #72253575, C3×#7918389
maximal quotient ofC3×#4421184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 33594);