Finite Group Databasebeta

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C7×#5745 #13061461

order1344
automorphism?
homologyAbelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25

Aliases

direct productC7×#5745

Splittings

FittingC2×C3×C7×M16 ⋊ C2
HallC3×C7 ⋊ #568
SylowC3 ⋊ C7×#568

Subgroups

subgroupG/H
centerC4×C7C23×D3
commutatorC2×C3C23×C4×C7
FrattiniC4C23×D3×C7
FittingC2×C3×C7×M16C2
socleC2×C3×C7C23×C4
prime radicalC2×C7×#235C2
cosocleC3×C4C24×C7
hypercenterC2×C7×M16D3
hypocenterC3C7×#568
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C2×C7×M16

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10750);