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C7×#5766 #13061482

order1344
automorphism?
homologyAbelianization (H1) = C24×C7 · Schur multiplier (H2) = C25

Aliases

direct productC7×#5766

Splittings

FittingC2×C3×C7×Q16 ⋊ C2
HallC3×C7 ⋊ #578
SylowC3 ⋊ C7×#578

Subgroups

subgroupG/H
centerC2×C7C2×D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC2×C3×C7×Q16C2
socleC2×C3×C7C22×D4
hypercenterC2×C7×Q16D3
hypocenterC3C7×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×Q16

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 42 points:

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10771);