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C3×#89894 #13055553

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

Aliases

direct productC3×#89894

Splittings

HallC3×C7 ⋊ #405
SylowC7 ⋊ C3×#405

Subgroups

subgroupG/H
centerC2×C3×C4C22×D7
commutatorC2×C7C2×C3×C42
FrattiniC2×C4C22×C3×D7
FittingC2×C3×C7×M16C2
socleC22×C3×C7C22×C4
prime radicalC23×C3×C7C2×C4
cosocleC2×C4×C7C23×C3
hypercenterC2×C3×M16D7
hypocenterC7C3×#405
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×M16

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

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, 4842);