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C2×C3×#2631 #10653192

order768
nilpotency class2
isoclinismisoclinic class of #2613
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×C3×#2631

Subgroups

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

3670 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);(8,11)(12,14)(15,16,18,19)(17,20,22,21);(4,6)(8,11)(12,14)(16,19)(20,21);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,19)(20,22);(7,9,10,13)(8,12,11,14)(15,17,18,22)(16,20,19,21)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×D3×#2631, C2×#11427644, #72270090, …
maximal subgroup ofC2×C3×#52004, C2×D3×#2631, C2×#11427644, …
maximal normal subgroup ofC2×C3×#52004, C2×D3×#2631, C2×#11427644, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 27573);