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Cayley diagram

C2×C22⋊ΓL(1,𝔽8) #4283

order168
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphismAut(G) = A4×AGL(1,𝔽7) (order 504) · Inn(G) = C22⋊ΓL(1,𝔽8) · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C22⋊ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊A4

Splittings

Fitting, socle, HallC23×C7 ⋊ C3
derivedC22×C7 ⋊ C2×C3
SylowC23 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C2×A4

Subgroups

subgroupG/H
center, hypercenterC2C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C2×C3
FrattiniC1G
Fitting, socleC23×C7C3
LCSG ▷ C22×C7
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic7⋊A4, C4×C22⋊ΓL(1,𝔽8), C2×D7⋊A4, …
maximal normal subgroup ofDic7⋊A4, C4×C22⋊ΓL(1,𝔽8), C2×D7⋊A4, …
maximal quotient ofC4×C22⋊ΓL(1,𝔽8), C2×C7⋊SL(2,𝔽3), #41⋊ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 53);