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Cayley diagram

(C22⋊D7)⋊C3 #4241

order168
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphismAut(G) = C22×AGL(1,𝔽7) (order 168) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C2
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(C22⋊D7)⋊C3

Splittings

FittingC22×C7 ⋊ C2×C3
HallC22⋊D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ D4
SylowC7 ⋊ C3×D4

Subgroups

subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C22×C3
FittingC22×C7C2×C3
hypercenterC22AGL(1,𝔽7)
hypocenterC7C3×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22

32 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:

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

Polycyclic presentation:

2^2,3,2,7|b^a=bd,e^a=e^6,e^c=e^4

Conjugacy classes & character table

19 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1157⋊C3, D4×AGL(1,𝔽7), #1159⋊C3, …
maximal normal subgroup of#1157⋊C3, D4×AGL(1,𝔽7), #1159⋊C3, …
maximal quotient of(ΓL(1,𝔽8)⋊C4)⋊C4, (C2×AGL(1,𝔽7))⋊C4, #1141⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 11);