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C3×#11680 #10630287

order768
nilpotency class4
isoclinismisoclinic class of #11677
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×#11680

Subgroups

subgroupG/H
center, socleC22×C3C2×#102
commutatorC2×C8C22×C3×C4
Frattini, cosocleC4×C8C23×C3
prime radicalC3×#3169C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G

368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#141800, C3×#141837, C3×#483961
maximal normal subgroup ofC3×#141800, C3×#141837, C3×#483961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4668);