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ΓL(1,𝔽8)×#103 #10620541

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

Aliases

direct productΓL(1,𝔽8)×#103
unambiguous semidirect product(C7×#103)⋊C3

Splittings

Fitting, HallC7×#103 ⋊ C3
SylowC7 ⋊ C3×#103

Subgroups

subgroupG/H
centerC22D4×ΓL(1,𝔽8)
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C22×ΓL(1,𝔽8)
FittingC7×#103C3
socleC22×C7C3×D4
prime radicalC22×ΓL(1,𝔽8)D4
cosocleC2×C4×C7C22×C3
hypercenter#103ΓL(1,𝔽8)
hypocenterC7C3×#103
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

84 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)(12,13,15,17)(14,19,18,16)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89590⋊C3, #89563⋊C3, #89564⋊C3, …
maximal normal subgroup of#89590⋊C3, #89563⋊C3, #89564⋊C3, …
maximal quotient ofΓL(1,𝔽8)×Q8⋊C8, ΓL(1,𝔽8)×#328, ΓL(1,𝔽8)×#332, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 123);