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C3×#26393 #10645000

order768
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#26393

Subgroups

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

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 768

A minimal faithful permutation representation — 4 generators on 31 points:

(1,2,3)(8,9,12,14)(10,13,11,15)(16,17,20,24)(18,25,26,30)(19,21,28,23)(22,29,31,27);(4,5)(6,7)(8,10,12,11)(9,13,14,15)(17,21)(18,26)(19,28)(22,30)(23,24)(25,31);(5,7)(8,11,12,10)(9,13,14,15);(16,18,20,26)(17,22,24,31)(19,29,28,27)(21,25,23,30)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 19381);