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C3×#17962 #10636569

order768
nilpotency class3
isoclinismisoclinic class of #3055
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC3×#17962

Subgroups

subgroupG/H
centerC2×C3×C4C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC42C24×C3
socleC22×C3C2×C4×D4
prime radicalC3×#3169C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C4×C8 ◁ G

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2,3)(4,5,7,9)(6,8,11,10)(12,13,16,20,17,14,21,19)(15,22,23,26,24,18,25,27);(5,8)(6,11)(9,10)(12,14)(13,17)(15,22)(16,19)(18,24)(20,21)(23,26)(25,27);(5,9)(8,10)(12,15,21,25,17,24,16,23)(13,18,19,26,14,22,20,27);(13,14)(18,22)(19,20)(26,27)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#213274, C3×#483961
maximal normal subgroup ofC3×#213274, C3×#483961
maximal quotient ofC3×#483961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 10950);