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Cayley diagram

C3×D5×D7 #88299

order420
isoclinismisoclinic class of D5×D7
automorphismAut(G) = C2×AGL(1,𝔽5)×AGL(1,𝔽7) (order 1680) · Inn(G) = D5×D7 · Out(G) = C22×C3
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×D5×D7

Splittings

Fitting, socle, HallC3×C5×C7 ⋊ C22
derived, HallC5×C7 ⋊ C22×C3
HallC3×C7 ⋊ C2×D5
HallC3×C5 ⋊ C2×D7
SylowC7 ⋊ C2×C3×D5
SylowC5 ⋊ C2×C3×D7

Subgroups

subgroupG/H
center, hypercenterC3D5×D7
commutator, cosocle, hypocenterC5×C7C22×C3
FrattiniC1G
Fitting, socleC3×C5×C7C22
LCSG ▷ C5×C7
UCSC1 ◁ C3

40 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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D7×AGL(1,𝔽5), D7×ΓL(1,𝔽16), D3×D5×D7, …
maximal normal subgroup ofC3×D7×AGL(1,𝔽5), D7×ΓL(1,𝔽16), D3×D5×D7, …
maximal quotient ofC3×D7×Dic5, C3×D5×Dic7, C3×#63304, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 24);