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Cayley diagram

C5×C7×D5 #66840

order350
isoclinismisoclinic class of D5
automorphismAut(G) = C2×C3×C4×AGL(1,𝔽5) (order 480) · Inn(G) = D5 · Out(G) = C22×C3×C4
homologyAbelianization (H1) = C2×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC5×C7×D5
unambiguous semidirect productC5⋊(C2×C5×C7)

Splittings

Fitting, socle, HallC52×C7 ⋊ C2
derivedC5 ⋊ C2×C5×C7
SylowC52 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC5×C7D5
commutator, cosocle, hypocenterC5C2×C5×C7
FrattiniC1G
Fitting, socleC52×C7C2
LCSG ▷ C5
UCSC1 ◁ C5×C7

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 70

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4,5)(7,8)(9,10)(11,12,13,14,15,16,17);(6,7,9,10,8)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×C7×AGL(1,𝔽5), C5×C7⋊AGL(1,𝔽5), C7×#1042, …
maximal normal subgroup ofC5×C7×AGL(1,𝔽5), C5×C7⋊AGL(1,𝔽5), C7×#1042, …
maximal quotient ofC5×C7×Dic5, C2×C5×C7×D5, C3×C5×C7×D5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(350, 6);