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ΓL(1,𝔽8)×#525 #13060686

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

Aliases

direct productΓL(1,𝔽8)×#525
unambiguous semidirect product(C7×#525)⋊C3

Splittings

Fitting, HallC7×#525 ⋊ C3
SylowC7 ⋊ C3×#525

Subgroups

subgroupG/H
centerC2×C4C23×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C7C24×C3
FrattiniC22C24×ΓL(1,𝔽8)
FittingC7×#525C3
hypercenter#525ΓL(1,𝔽8)
hypocenterC7C3×#525
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #525

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 4 generators on 19 points:

(2,3,4)(5,7,6)(8,9)(10,11)(12,13)(14,17)(15,16)(18,19);(1,2,3,5,4,6,7)(13,16)(17,19);(9,11);(12,14,15,18)(13,17,16,19)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#525, ΓL(1,𝔽8)×#3484, #72229268, …
maximal normal subgroup ofAGL(1,𝔽7)×#525, ΓL(1,𝔽8)×#3484, #72229268, …
maximal quotient ofC4×ΓL(1,𝔽8)×#120

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 9975);