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C3×#24203 #10642810

order768
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#24203

Subgroups

subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

1036 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)(4,5)(6,7)(8,9)(10,14)(11,12)(13,15)(16,17,22,18)(19,27,28,31)(20,25,21,24)(23,30,26,29);(9,12)(13,14)(16,18,22,17)(19,27,28,31)(20,24,21,25)(23,30,26,29);(5,7)(16,19,22,28)(17,23,18,26)(20,30,21,29)(24,31,25,27);(8,10,11,15)(9,13,12,14)(19,28)(23,26)(27,31)(29,30)

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, 17191);