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C3×#37129 #10655736

order768
nilpotency class2
isoclinismisoclinic class of #36702
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#37129

Subgroups

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

2890 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)(8,9)(10,12)(11,14)(13,15)(17,21)(19,23);(9,10)(13,14)(17,21)(18,22);(4,5,6,7)(16,17,20,21)(18,19,22,23);(8,10,12,9)(11,14,15,13)(16,18,20,22)(17,19,21,23);(8,11)(9,13)(10,14)(12,15)(16,19)(17,22)(18,21)(20,23)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270719, D3×#37129
maximal subgroup of#72270719, D3×#37129, C3×#7659307
maximal normal subgroup of#72270719, D3×#37129, C3×#7659307
maximal quotient ofC3×#763332

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 30117);