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ΓL(1,𝔽8)×#99 #10620537

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productΓL(1,𝔽8)×#99
unambiguous semidirect product(C7×#99)⋊C3

Splittings

Fitting, HallC7×#99 ⋊ C3
SylowC7 ⋊ C3×#99

Subgroups

subgroupG/H
centerC2C22⋊C4×ΓL(1,𝔽8)
commutatorC22×C7C2×C3×C4
FrattiniC23C22×ΓL(1,𝔽8)
FittingC7×#99C3
socleC2×C7C3×C22⋊C4
prime radicalC2×D4×ΓL(1,𝔽8)C2
cosocleC23×C7C22×C3
hypercenter#99ΓL(1,𝔽8)
hypocenterC7C3×#99
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C23 ◁ #99

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89266⋊C3, #89267⋊C3, #89268⋊C3, …
maximal normal subgroup of#89266⋊C3, #89267⋊C3, #89268⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#323, ΓL(1,𝔽8)×#324, ΓL(1,𝔽8)×#327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 119);