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C3×#89288 #13051153

order1344
isoclinismisoclinic class of #6400
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#89288

Splittings

HallC3×C7 ⋊ #368
SylowC7 ⋊ C3×#368

Subgroups

subgroupG/H
centerC3×C4#1141
commutatorC7×C8C2×C3×C4
FrattiniC2×C8C2×C3×D7
FittingC3×C7×#108C2
socleC2×C3×C7#107
prime radicalC22×C3×C7Q16
cosocleC2×C7×C8C22×C3
hypercenterC3×#108D7
hypocenterC7C3×#368
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C3×#108

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3)(4,5,7,9,10,8,6)(11,13,14,22,16,25,27,24)(12,17,18,30,20,32,34,31)(15,28,26,23,29,39,41,40)(19,35,33,21,36,37,42,38);(5,6)(7,8)(9,10)(11,12,15,19,27,34,41,42,16,20,29,36,14,18,26,33)(13,21,23,30,22,37,39,32,25,38,40,31,24,35,28,17)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 442);