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C3×#20989 #10639596

order768
nilpotency class3
isoclinismisoclinic class of #1468
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC3×#20989

Subgroups

subgroupG/H
centerC2×C3×C4C2×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC22×C4C24×C3
socleC22×C3C22×C22⋊C4
prime radicalC2×C3×#525C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C24×C3×C4 ◁ G

1874 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)(8,9,12,17,13,16,11,15)(10,14,18,21,19,22,20,23);(8,10,12,20,13,19,11,18)(9,14,15,21,16,22,17,23);(5,7);(8,11,13,12)(9,15,16,17)(10,18,19,20)(14,21,22,23)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#20989, #72249240, #72253860, …
maximal subgroup ofD3×#20989, C3×#359639, #72249240, …
maximal normal subgroup ofD3×#20989, C3×#359639, #72249240, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13977);