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C2×C3×#3291 #10679709

order768
nilpotency class3
isoclinismisoclinic class of #3290
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×C3×#3291

Subgroups

subgroupG/H
center, socleC22×C3D42
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

2682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 21 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×#61110
maximal normal subgroup ofC2×C3×#61110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 54090);