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C3×#33517 #10652124

order768
nilpotency class4
isoclinismisoclinic class of #33517
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#33517

Subgroups

subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#33517, C3×#495725, #72219832
maximal normal subgroup ofD3×#33517, C3×#495725, #72219832

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26505);