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Cayley diagram

Q8×ΓL(1,𝔽8) #4251

order168
isoclinismisoclinic class of D4×ΓL(1,𝔽8)
automorphismAut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = C22×ΓL(1,𝔽8) · Out(G) = C2×D3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productQ8×ΓL(1,𝔽8)

Splittings

Fitting, HallC7×Q8 ⋊ C3
SylowC7 ⋊ C3×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×ΓL(1,𝔽8)
commutator, socle, cosocleC2×C7C22×C3
FittingC7×Q8C3
prime radicalC2×ΓL(1,𝔽8)C22
hypercenterQ8ΓL(1,𝔽8)
hypocenterC7C3×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ Q8

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:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1143⋊C3, #1144⋊C3, SD16×ΓL(1,𝔽8), …
maximal normal subgroup of#1143⋊C3, #1144⋊C3, SD16×ΓL(1,𝔽8), …
maximal quotient ofC4⋊C4×ΓL(1,𝔽8), C2×Q8×ΓL(1,𝔽8), Q8×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 21);