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Cayley diagram

C4×ΓL(1,𝔽8) #743

order84
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C2×AGL(1,𝔽7) (order 84) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22
holomorphG ⋊ Aut(G) = D4×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC4ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4
FrattiniC2C2×ΓL(1,𝔽8)
FittingC4×C7C3
socle, cosocleC2×C7C2×C3
prime radicalC2×ΓL(1,𝔽8)C2
LCSG ▷ C7
UCSC1 ◁ C4

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

(2,3,4)(5,7,6)(8,9,10,11);(1,2,3,5,4,6,7)

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊C8, C8×ΓL(1,𝔽8), ΓL(1,𝔽8)⋊Q8, …
maximal normal subgroup ofΓL(1,𝔽8)⋊C8, C8×ΓL(1,𝔽8), ΓL(1,𝔽8)⋊Q8, …
maximal quotient ofC8×ΓL(1,𝔽8), C2×C4×ΓL(1,𝔽8), C4×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 2);