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Cayley diagram

C3×C7×D5 #6147

order210
isoclinismisoclinic class of D5
automorphismAut(G) = C22×C3×AGL(1,𝔽5) (order 240) · Inn(G) = D5 · Out(G) = C23×C3
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC3×C7×D5
unambiguous semidirect productC5⋊(C2×C3×C7)

Splittings

Fitting, socle, HallC3×C5×C7 ⋊ C2
derived, SylowC5 ⋊ C2×C3×C7
HallC5×C7 ⋊ C2×C3
HallC3×C5 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC3×C7D5
commutator, cosocle, hypocenterC5C2×C3×C7
FrattiniC1G
Fitting, socleC3×C5×C7C2
LCSG ▷ C5
UCSC1 ◁ C3×C7

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 105

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C7×AGL(1,𝔽5), C3×C7⋊AGL(1,𝔽5), C7×ΓL(1,𝔽16), …
maximal normal subgroup ofC3×C7×AGL(1,𝔽5), C3×C7⋊AGL(1,𝔽5), C7×ΓL(1,𝔽16), …
maximal quotient ofC3×C7×Dic5, C2×C3×C7×D5, C7×C9×D5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(210, 6);