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C3×#89769 #13055429

order1344
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×#89769

Splittings

HallC3×C7 ⋊ #405
SylowC7 ⋊ C3×#405

Subgroups

subgroupG/H
centerC2×C3×C4C22×D7
commutatorC2×C7C2×C3×C42
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#117C2
socleC22×C3×C7C22×C4
prime radicalC23×C3×C7C2×C4
cosocleC2×C4×C7C23×C3
hypercenterC3×#117D7
hypocenterC7C3×#405
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3×C4 ◁ C3×#117

260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 672

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(4,5,7,9,10,8,6)(11,12,13,14)(17,23)(20,26)(24,29)(27,30);(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(5,6)(7,8)(9,10)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30)

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4718);