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Cayley diagram

D3×D35 #88304

order420
isoclinismisoclinic class of D3×D35
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productD3×D35
unambiguous semidirect product(C5×D3)⋊D7
unambiguous semidirect product(D3×C7)⋊D5

Splittings

Fitting, socle, derived, HallC3×C5×C7 ⋊ C22
HallC5×C7 ⋊ C2×D3
HallC3×C7 ⋊ C2×D5
HallC3×C5 ⋊ C2×D7
SylowC7 ⋊ D3×D5
SylowC5 ⋊ D3×D7
SylowC3 ⋊ C2×D35

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7C22
LCSG ▷ C3×C5×C7
UCSC1

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D5×D7, C2×D3×D35, D9×D35, …
maximal normal subgroup ofD3×D5×D7, C2×D3×D35, D3×C5⋊AGL(1,𝔽7), …
maximal quotient ofDic3×D35, D3×Dic35, #11719777, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 29);