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C3×#89499 #13055161

order1344
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×#89499

Splittings

FittingC3×C7×C22⋊C8 ⋊ C2
HallC3×C7 ⋊ #408
SylowC7 ⋊ C3×#408

Subgroups

subgroupG/H
centerC2×C3×C4C22×D7
commutatorC22×C7C22×C3×C4
FrattiniC2×C4C22×C3×D7
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C22×C4
prime radicalC2×C3×#1157C2
cosocleC2×C4×C7C23×C3
hypercenterC3×C22⋊C8D7
hypocenterC7C3×#408
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

316 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)(12,14)(15,17,18,23,19,24,25,29)(16,21,22,28,20,26,27,30);(11,12)(13,14)(16,20)(21,26)(22,27)(28,30);(5,6)(7,8)(9,10)(15,16)(17,21)(18,22)(19,20)(23,28)(24,26)(25,27)(29,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4450);