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Cayley diagram

ΓL(1,𝔽8)⋊Q8 #4237

order168
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphismAut(G) = D4×AGL(1,𝔽7) (order 336) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊Q8
unambiguous semidirect productDic14⋊C3

Splittings

HallDic14 ⋊ C3
HallΓL(1,𝔽8) ⋊ Q8
SylowC7 ⋊ C3×Q8

Subgroups

subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽7)
commutator, socle, cosocleC2×C7C22×C3
FittingC4×C7C2×C3
prime radicalC2×ΓL(1,𝔽8)C22
hypercenterC4AGL(1,𝔽7)
hypocenterC7C3×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C4

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2^2,3,2,7|a^2=d,b^2=d,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#1132⋊C3, Dic28⋊C3, #1142⋊C3, …
maximal normal subgroup of#1132⋊C3, Dic28⋊C3, #1142⋊C3, …
maximal quotient of(ΓL(1,𝔽8)⋊C4)⋊C4, C4⋊(ΓL(1,𝔽8)⋊C4), C2×ΓL(1,𝔽8)⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 7);