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ΓL(1,𝔽8)×#1910 #72277378

order2688
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4

Aliases

direct productΓL(1,𝔽8)×#1910

Splittings

Fitting, HallC7×#1910 ⋊ C3
SylowC7 ⋊ C3×#1910

Subgroups

subgroupG/H
centerC2ΓL(1,𝔽8)×#386
commutatorC2×C4×C7C22×C3×C4
FrattiniC22×C4C23×ΓL(1,𝔽8)
FittingC7×#1910C3
socleC2×C7C3×#386
prime radicalΓL(1,𝔽8)×#535C2
cosocleC22×C4×C7C23×C3
hypercenter#1910ΓL(1,𝔽8)
hypocenterC7C3×#1910
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1910

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#1910
maximal normal subgroup ofAGL(1,𝔽7)×#1910

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,4)(5,7,6)(10,16)(11,18)(13,19)(14,22)(17,23)(20,21), (1,2,4,6,3,5,7)(8,10,11,16,12,17,18,23)(9,13,14,20,15,21,22,19), (8,9)(10,13)(11,14)(12,15)(16,20)(17,21)(18,22)(19,23));