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Cayley diagram

C5×C7×A4 #88307

order420
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C3×C4×S4 (order 576) · Inn(G) = A4 · Out(G) = C22×C3×C4
homologyAbelianization (H1) = C3×C5×C7 · Schur multiplier (H2) = C2

Aliases

direct productC5×C7×A4
unambiguous semidirect productC22⋊(C3×C5×C7)
unambiguous semidirect product(C22×C5)⋊(C3×C7)

Splittings

Fitting, socle, HallC22×C5×C7 ⋊ C3
derived, SylowC22 ⋊ C3×C5×C7
HallC22×C7 ⋊ C3×C5
HallC22×C5 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC5×C7A4
commutator, cosocle, hypocenterC22C3×C5×C7
FrattiniC1G
Fitting, socleC22×C5×C7C3
LCSG ▷ C22
UCSC1 ◁ C5×C7

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 140

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3,4)(5,6,7,8,9)(10,11,12,13,14,15,16);(1,4)(2,3)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊(C7⋊S4), C7⋊(A4⋊Dic5), C7⋊(A4⋊AGL(1,𝔽5)), …
maximal subgroup ofC5×C7×S4, C5×C7⋊S4, C7×C5⋊S4, …
maximal normal subgroup ofC5×C7×S4, C5×C7⋊S4, C7×C5⋊S4, …
maximal quotient ofC5×C7×SL(2,𝔽3), C2×C5×C7×A4, C5×C7×C22⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 32);