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Cayley diagram

C22×C3×AGL(1,𝔽7) #92752

order504
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC22×C3×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C22×C32)
unambiguous semidirect productΓL(1,𝔽8)⋊(C23×C3)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C22×C3)

Splittings

Fitting, socleC22×C3×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C23×C32
HallC3×ΓL(1,𝔽8) ⋊ C23
HallC22×D7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC22×C3AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C23×C32
FrattiniC1G
Fitting, socleC22×C3×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C22×C3

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

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

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽7))⋊Dic3, C3×(C2×AGL(1,𝔽7))⋊C4, C2×Dic3×AGL(1,𝔽7), …
maximal normal subgroup of(C2×AGL(1,𝔽7))⋊Dic3, C3×(C2×AGL(1,𝔽7))⋊C4, C2×Dic3×AGL(1,𝔽7), …
maximal quotient ofC2×C3×ΓL(1,𝔽8)⋊Q8, C2×C3×C4×AGL(1,𝔽7), C2×C3×C4⋊AGL(1,𝔽7), …
automorphism group ofC7×Dic7, C2×C7×D7, C9×Dic7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 178);