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ΓL(1,𝔽8)×#401 #13054932

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

Aliases

direct productΓL(1,𝔽8)×#401

Splittings

Fitting, HallC7×#401 ⋊ C3
SylowC7 ⋊ C3×#401

Subgroups

subgroupG/H
center, FrattiniC23C23×ΓL(1,𝔽8)
commutator, socle, cosocleC23×C7C23×C3
FittingC7×#401C3
prime radicalC23×ΓL(1,𝔽8)C23
hypercenter#401ΓL(1,𝔽8)
hypocenterC7C3×#401
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #401

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

(2,3,4)(5,7,6)(10,14)(12,15)(17,20)(21,22)(24,25,27,29)(26,30,31,28);(1,2,3,5,4,6,7)(8,9,11,13)(10,15,14,12)(16,17,19,20)(18,22,23,21)(24,25)(26,30)(27,29)(28,31);(8,10)(9,12)(11,14)(13,15)(16,18,19,23)(17,21,20,22)(24,26,27,31)(25,28,29,30)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4221);