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C3×#29314 #10647921

order768
nilpotency class3
isoclinismisoclinic class of #24378
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C23

Aliases

direct productC3×#29314

Subgroups

subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 768

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

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 22302);