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C2×C3×#3288 #10679705

order768
nilpotency class3
isoclinismisoclinic class of #3288
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×C3×#3288

Subgroups

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

2704 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);(4,9)(6,7)(10,17)(11,15)(12,18)(13,16);(3,4)(5,12)(6,15)(7,9)(8,11)(10,13)(14,17)(16,18);(3,4)(5,10)(6,9)(7,15)(8,11)(12,13)(14,18)(16,17);(3,5)(4,10)(6,16)(7,13)(8,14)(9,17)(11,18)(12,15)

Conjugacy classes & character table

156 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, 54086);