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Cayley diagram

Q8×AGL(1,𝔽8) #90602

order448
isoclinismisoclinic class of D4×AGL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C1

Aliases

direct productQ8×AGL(1,𝔽8)
unambiguous semidirect product(C23×Q8)⋊C7

Splittings

Fitting, SylowC23×Q8 ⋊ C7

Subgroups

subgroupG/H
center, FrattiniC2C22×AGL(1,𝔽8)
commutator, socle, cosocleC24C22×C7
FittingC23×Q8C7
prime radicalC2×AGL(1,𝔽8)C22
hypercenterQ8AGL(1,𝔽8)
hypocenterC23C7×Q8
LCSG ▷ C24 ▷ C23
UCSC1 ◁ C2 ◁ Q8

83 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 16 points:

(1,3,4,8)(2,5,6,7)(10,11,12,14,16,13,15);(1,2,4,6)(3,7,8,5)(9,13)(10,11)(12,16)(14,15)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup ofQ8⋊AΓL(1,𝔽8)
maximal subgroup ofSD16×AGL(1,𝔽8), Q16×AGL(1,𝔽8), C2×Q8×AGL(1,𝔽8), …
maximal normal subgroup ofSD16×AGL(1,𝔽8), Q16×AGL(1,𝔽8), C2×Q8×AGL(1,𝔽8), …
maximal quotient ofC4⋊C4×AGL(1,𝔽8), C2×Q8×AGL(1,𝔽8), Dic6×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1364);