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C4×C8×ΓL(1,𝔽8) #10620534

order672
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C4

Aliases

direct productC4×C8×ΓL(1,𝔽8)
unambiguous semidirect product(C4×C7×C8)⋊C3

Splittings

Fitting, HallC4×C7×C8 ⋊ C3
derived, SylowC7 ⋊ C3×C4×C8

Subgroups

subgroupG/H
center, hypercenterC4×C8ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C8
FrattiniC2×C4C22×ΓL(1,𝔽8)
FittingC4×C7×C8C3
socleC22×C7C2×C3×C4
prime radicalC22×ΓL(1,𝔽8)C2×C4
cosocleC2×C4×C7C22×C3
LCSG ▷ C7
UCSC1 ◁ C4×C8

88 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×ΓL(1,𝔽8)⋊C8, #89249⋊C3, #89250⋊C3, …
maximal normal subgroup ofC8×ΓL(1,𝔽8)⋊C8, #89249⋊C3, #89250⋊C3, …
maximal quotient ofC82×ΓL(1,𝔽8), ΓL(1,𝔽8)×#322, ΓL(1,𝔽8)×#336, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 116);