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Cayley diagram

C2×ΓL(1,𝔽8)⋊C4 #4240

order168
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = D4×AGL(1,𝔽7) (order 336) · Inn(G) = AGL(1,𝔽7) · Out(G) = D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)⋊C4
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊C4
unambiguous semidirect product(C2×Dic7)⋊C3

Splittings

derived, SylowC7 ⋊ C2×C3×C4
HallC2×Dic7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C4
FrattiniC2C2×AGL(1,𝔽7)
Fitting, socleC22×C7C2×C3
prime radicalC22×ΓL(1,𝔽8)C2
cosocleC2×C7C22×C3
LCSG ▷ C7
UCSC1 ◁ C22

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 56

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

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×ΓL(1,𝔽8)⋊C4, (ΓL(1,𝔽8)⋊C4)⋊C4, C4⋊(ΓL(1,𝔽8)⋊C4), …
maximal normal subgroup ofC4×ΓL(1,𝔽8)⋊C4, (ΓL(1,𝔽8)⋊C4)⋊C4, C4⋊(ΓL(1,𝔽8)⋊C4), …
maximal quotient ofC2×ΓL(1,𝔽8)⋊C8, #1136⋊C3, C4×ΓL(1,𝔽8)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 10);