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C3×#11041 #10629648

order768
nilpotency class3
isoclinismisoclinic class of #2010
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC3×#11041

Subgroups

subgroupG/H
centerC2×C3×C4#127
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
socleC22×C3#392
prime radicalC2×C3×#575C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G

1374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#11041, #72271667, #72271779
maximal subgroup ofC3×#394022, D3×#11041, #72271667, …
maximal normal subgroup ofC3×#394022, D3×#11041, #72271667, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4029);