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Cayley diagram

C7×A5 #88288

order420
isoclinismisoclinic class of A5
automorphismAut(G) = C2×C3×S5 (order 720) · Inn(G) = A5 · Out(G) = C22×C3
homologyAbelianization (H1) = C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×A5

Subgroups

subgroupG/H
center, Fitting, hypercenterC7A5
commutator, hypocenterA5C7
FrattiniC1G
LCSG ▷ A5
UCSC1 ◁ C7

18 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:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC7⋊S5, D7×A5, ΓL(1,𝔽8)×A5, …
socle ofC7×S5, C7⋊S5, D7×A5, …
maximal subgroup ofC7×S5, C7⋊S5, D7×A5, …
maximal normal subgroup ofC7×S5, C7⋊S5, D7×A5, …
maximal quotient ofC7×SL(2,𝔽5), C2×C7×A5, C3×C7×A5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 13);