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C3×#37744 #10656351

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

Aliases

direct productC3×#37744

Subgroups

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

3176 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)(8,9)(10,14)(11,12)(13,15)(16,17)(18,19)(20,21)(22,23);(5,7)(9,12)(13,14)(17,21)(19,22);(4,5)(6,7)(16,18)(17,22)(19,21)(20,23);(8,10)(9,13)(11,15)(12,14)(16,18)(17,22)(19,21)(20,23);(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,19,20,22)(17,18,21,23)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72247956, D3×#37744
maximal subgroup of#72247956, D3×#37744, C3×#7849701
maximal normal subgroup of#72247956, D3×#37744, C3×#7849701
maximal quotient ofC3×#850465, C3×#1085256

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 30732);