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C3×#60688 #10679295

order768
nilpotency class3
isoclinismisoclinic class of #3055
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#60688

Subgroups

subgroupG/H
centerC2×C3×C4C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
socleC22×C3C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 5 generators on 27 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#10576928, D3×#60688, #72225908
maximal normal subgroup ofC3×#10576928, D3×#60688, #72225908

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53676);