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Cayley diagram

D7×AGL(1,𝔽5) #63327

order280
isoclinismisoclinic class of D7×AGL(1,𝔽5)
automorphismAut(G) = AGL(1,𝔽5)×AGL(1,𝔽7) (order 840) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productD7×AGL(1,𝔽5)
unambiguous semidirect product(C7×AGL(1,𝔽5))⋊C2
unambiguous semidirect product(C7⋊AGL(1,𝔽5))⋊C2

Splittings

Fitting, socle, derived, HallC5×C7 ⋊ C2×C4
SylowC7 ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C4×D7

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7C2×C4
prime radicalD5×D7C2
cosocleC7×D5C22
LCSG ▷ C5×C7
UCSC1

32 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:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D7×AGL(1,𝔽5), AGL(1,𝔽5)×AGL(1,𝔽7), C3×D7×AGL(1,𝔽5), …
maximal normal subgroup ofC2×D7×AGL(1,𝔽5), AGL(1,𝔽5)×AGL(1,𝔽7), C3×D7×AGL(1,𝔽5), …
maximal quotient ofAGL(1,𝔽5)×Dic7, (C2×D7)⋊AGL(1,𝔽5), Dic7⋊AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(280, 32);