Finite Group Databasebeta

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C2×C3×#2688 #10670715

order768
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×C3×#2688

Subgroups

subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

3650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 5 generators on 25 points:

(1,2)(23,24,25);(7,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,22)(19,21);(4,6)(8,9)(10,14)(15,17,19,22)(16,20,21,18);(3,4)(5,6)(7,9,11,8)(10,13,14,12)(15,17)(16,18)(19,22)(20,21);(7,10)(8,12)(9,13)(11,14)(15,18)(16,17)(19,20)(21,22)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#11427639, C2×D3×#2688, #72270100, …
maximal subgroup ofC2×C3×#52004, C2×#11427639, C2×D3×#2688, …
maximal normal subgroup ofC2×C3×#52004, C2×#11427639, C2×D3×#2688, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 45096);