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C3×#35840 #10654447

order768
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#35840

Subgroups

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

2726 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)(4,5,6,7)(18,22)(21,23);(4,5)(6,7)(8,9)(10,13)(11,14)(12,15)(16,17,20,19)(18,21,22,23);(4,5)(6,7)(8,10)(9,12)(11,15)(13,14)(16,18)(17,21)(19,23)(20,22);(8,10)(9,13)(11,15)(12,14)(16,19,20,17)(18,23,22,21);(8,9)(10,12)(11,14)(13,15)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#35840, #72253526
maximal subgroup ofD3×#35840, #72253526, C3×#7918389
maximal normal subgroup ofD3×#35840, #72253526, C3×#7918389
maximal quotient ofC3×#4929019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 28828);