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ΓL(1,𝔽8)×#106 #10620544

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×#106
unambiguous semidirect product(C7×#106)⋊C3

Splittings

Fitting, HallC7×#106 ⋊ C3
SylowC7 ⋊ C3×#106

Subgroups

subgroupG/H
centerC22D4×ΓL(1,𝔽8)
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C22×ΓL(1,𝔽8)
FittingC7×#106C3
socleC22×C7C3×D4
prime radicalC22×ΓL(1,𝔽8)D4
cosocleC2×C4×C7C22×C3
hypercenter#106ΓL(1,𝔽8)
hypocenterC7C3×#106
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89282⋊C3, #89283⋊C3, ΓL(1,𝔽8)×#360, …
maximal normal subgroup of#89282⋊C3, #89283⋊C3, ΓL(1,𝔽8)×#360, …
maximal quotient ofΓL(1,𝔽8)×#334, ΓL(1,𝔽8)×#340, ΓL(1,𝔽8)×#365, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 126);