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C2×C3×#3109 #10679148

order768
nilpotency class3
isoclinismisoclinic class of #3003
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×C3×#3109

Subgroups

subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

2830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 192

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

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×D3×#3109, #72269993
maximal subgroup ofC2×D3×#3109, #72269993, C3×#6453969
maximal normal subgroup ofC2×D3×#3109, #72269993, C3×#6453969
maximal quotient ofC3×#7623148

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53529);