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Cayley diagram

C2×C5⋊AGL(1,𝔽7) #88294

order420
isoclinismisoclinic class of C5⋊AGL(1,𝔽7)
automorphismAut(G) = C2×AGL(1,𝔽5)×AGL(1,𝔽7) (order 1680) · Inn(G) = C5⋊AGL(1,𝔽7) · Out(G) = C2×C4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5⋊AGL(1,𝔽7)
unambiguous semidirect product(C2×C5)⋊AGL(1,𝔽7)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D5
unambiguous semidirect product(C2×D35)⋊C3

Splittings

Fitting, socleC2×C5×C7 ⋊ C2×C3
derived, HallC5×C7 ⋊ C22×C3
HallC2×D35 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C22
HallΓL(1,𝔽8) ⋊ C2×D5
SylowC7 ⋊ C2×C3×D5
SylowC5 ⋊ C2×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2C5⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC5×C7C22×C3
FrattiniC1G
Fitting, socleC2×C5×C7C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 70

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#63304⋊C3, #63306⋊C3, #63307⋊C3, …
maximal normal subgroup of#63304⋊C3, #63306⋊C3, #63307⋊C3, …
maximal quotient ofDic70⋊C3, C4×C5⋊AGL(1,𝔽7), C4⋊(C5⋊AGL(1,𝔽7)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 19);