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C2×C3×#3027 #10679003

order768
nilpotency class3
isoclinismisoclinic class of #3026
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×C3×#3027

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

5110 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);(6,13)(8,14)(9,16)(11,17);(3,4)(5,8)(6,9)(7,10)(11,15)(12,14)(13,16)(17,18);(3,5)(4,8)(6,11)(7,12)(9,15)(10,14)(13,17)(16,18);(3,6,7,13)(4,9,10,16)(5,11,12,17)(8,15,14,18)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×D3×#3027, C2×#10949264, #72269875, …
maximal subgroup ofC2×C3×#33553, C3×#498888, C3×#394076, …
maximal normal subgroup ofC2×C3×#33553, C3×#498888, C3×#394076, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53384);