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C3×#89312 #13051177

order1344
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#89312

Splittings

HallC3×C7 ⋊ #362
SylowC7 ⋊ C3×#362

Subgroups

subgroupG/H
centerC2×C3#6395
commutatorC7×C8C2×C3×C4
FrattiniC2×C8C2×C3×D7
FittingC3×C7×M32C2
socleC2×C3×C7#102
prime radicalC22×C3×C7SD16
cosocleC2×C7×C8C22×C3
hypercenterC3×M32D7
hypocenterC7C3×#362
LCSG ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C3×M32

108 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)(11,13,14,26,27,38,33,39,16,23,29,22,15,24,28,42)(12,17,18,34,30,36,41,37,20,31,35,21,19,32,25,40);(5,6)(7,8)(9,10)(11,12,16,20)(13,21,23,34)(14,25,29,41)(15,30,27,19)(17,26,31,22)(18,33,35,28)(24,40,38,37)(32,39,36,42)

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 466);