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ΓL(1,𝔽8)×#519 #13060680

order1344
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productΓL(1,𝔽8)×#519

Splittings

Fitting, HallC7×#519 ⋊ C3
SylowC7 ⋊ C3×#519

Subgroups

subgroupG/H
center, FrattiniC22C24×ΓL(1,𝔽8)
commutatorC2×C7C23×C3×C4
FittingC7×#519C3
socle, cosocleC22×C7C24×C3
prime radicalC23×ΓL(1,𝔽8)C23
hypercenter#519ΓL(1,𝔽8)
hypocenterC7C3×#519
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #519

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 9969);