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Cayley diagram

C23×AGL(1,𝔽8) #90630

order448
isoclinismisoclinic class of AGL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C23

Aliases

direct productC23×AGL(1,𝔽8)

Splittings

Fitting, socle, SylowC26 ⋊ C7
derivedC23 ⋊ C23×C7

Subgroups

subgroupG/H
center, hypercenterC23AGL(1,𝔽8)
commutator, cosocle, hypocenterC23C23×C7
FrattiniC1G
Fitting, socleC26C7
LCSG ▷ C23
UCSC1 ◁ C23

463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

derived subgroup ofAGL(1,𝔽8)×AΓL(1,𝔽8)
maximal subgroup ofC22⋊C4×AGL(1,𝔽8), C22×C4×AGL(1,𝔽8), C2×D4×AGL(1,𝔽8), …
maximal normal subgroup ofC22⋊C4×AGL(1,𝔽8), C22×C4×AGL(1,𝔽8), C2×D4×AGL(1,𝔽8), …
maximal quotient ofC22×C4×AGL(1,𝔽8), C2×D4×AGL(1,𝔽8), C2×Q8×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1392);