Finite Group Databasebeta

← Prev (#87658) Next (#87660) →

Cayley diagram

C7×AGL(1,𝔽8) #87659

order392
isoclinismisoclinic class of AGL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C72 · Schur multiplier (H2) = C7

Aliases

BorelB(2,𝔽8)
direct productC7×AGL(1,𝔽8)
unambiguous semidirect productC23⋊C72
unambiguous semidirect product(C23×C7)⋊C7

Splittings

Fitting, socleC23×C7 ⋊ C7
derived, SylowC23 ⋊ C72

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽8)
commutator, cosocle, hypocenterC23C72
FrattiniC1G
Fitting, socleC23×C7C7
LCSG ▷ C23
UCSC1 ◁ C7

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:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11924305, #11924306, C2×#11924306, …
maximal subgroup ofD7×AGL(1,𝔽8), C2×C7×AGL(1,𝔽8), C7×AΓL(1,𝔽8), …
maximal normal subgroup ofD7×AGL(1,𝔽8), C2×C7×AGL(1,𝔽8), C7×AΓL(1,𝔽8), …
maximal quotient ofC2×C7×AGL(1,𝔽8), C3×C7×AGL(1,𝔽8), C5×C7×AGL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(392, 39);