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C3×#61050 #10679657

order768
nilpotency class3
isoclinismisoclinic class of #61049
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC3×#61050

Subgroups

subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#525 ◁ G

2870 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)(4,5)(6,7)(8,9)(10,16)(11,20)(12,14)(13,18)(15,21)(17,23)(19,22);(8,10,12,19)(9,13,14,23)(11,17,21,18)(15,22,20,16);(5,7);(10,17)(11,21)(13,22)(15,20)(16,23)(18,19);(9,14)(13,23)(15,20)(16,22)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253862, D3×#61050
maximal subgroup of#72253862, D3×#61050, C3×#10576108
maximal normal subgroup of#72253862, D3×#61050, C3×#10576108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 54038);