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Cayley diagram

C7×AGL(1,𝔽5) #3647

order140
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×C3×AGL(1,𝔽5) (order 120) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2×C3
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)2×AGL(1,𝔽7)
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×AGL(1,𝔽5)

Splittings

Fitting, socle, HallC5×C7 ⋊ C4
derived, SylowC5 ⋊ C4×C7

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽5)
commutator, hypocenterC5C4×C7
FrattiniC1G
Fitting, socleC5×C7C4
prime radicalC7×D5C2
cosocleD5C2×C7
LCSG ▷ C5
UCSC1 ◁ C7

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 35

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×AGL(1,𝔽5), C2×C7×AGL(1,𝔽5), AGL(1,𝔽5)×ΓL(1,𝔽8), …
maximal normal subgroup ofD7×AGL(1,𝔽5), C2×C7×AGL(1,𝔽5), AGL(1,𝔽5)×ΓL(1,𝔽8), …
maximal quotient ofC7×#170, C2×C7×AGL(1,𝔽5), C3×C7×AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(140, 5);