Finite Group Databasebeta

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C7×#5122 #13056155

order1344
automorphism?
homologyAbelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4

Aliases

direct productC7×#5122

Splittings

HallC3×C7 ⋊ #405
SylowC3 ⋊ C7×#405

Subgroups

subgroupG/H
centerC2×C4×C7C22×D3
commutatorC2×C3C2×C42×C7
FrattiniC2×C4C22×D3×C7
FittingC2×C3×C7×M16C2
socleC22×C3×C7C22×C4
prime radicalC23×C3×C7C2×C4
cosocleC2×C3×C4C23×C7
hypercenterC2×C7×M16D3
hypocenterC3C7×#405
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4×C7 ◁ C2×C7×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:

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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 5444);