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C2×C3×#2213 #10652199

order768
nilpotency class4
isoclinismisoclinic class of #2212
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC2×C3×#2213

Subgroups

subgroupG/H
center, socleC22×C3#459
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×#2213, C3×#495850, #72226023, …
maximal normal subgroup ofC2×D3×#2213, C3×#495850, #72226023, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26580);