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C3×#90516 #13061162

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#90516

Splittings

HallC3×C7 ⋊ #536
SylowC7 ⋊ C3×#536

Subgroups

subgroupG/H
centerC22×C3C23×D7
commutator, cosocleC22×C7C24×C3
FrattiniC22C23×C3×D7
Fitting, prime radicalC2×C3×C7×#41C2
socleC22×C3×C7C24
hypercenterC2×C3×#41D7
hypocenterC7C3×#536
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×#41

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10451);