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Cayley diagram

C3×Dic35 #88282

order420
isoclinismisoclinic class of D35
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×Dic35

Splittings

derived, HallC5×C7 ⋊ C3×C4
HallC3×C5×C7 ⋊ C4
HallC3×C7 ⋊ Dic5
HallC3×C5 ⋊ Dic7
SylowC7 ⋊ C3×Dic5
SylowC5 ⋊ C3×Dic7

Subgroups

subgroupG/H
center, hypercenterC2×C3D35
commutator, hypocenterC5×C7C3×C4
FrattiniC2C3×D35
Fitting, socle, prime radicalC2×C3×C5×C7C2
cosocleC2×C5×C7C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C2×C3

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 420

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3)(4,5)(6,7)(8,9,10,11,12)(13,14,15,16,17,18,19);(4,6,5,7)(9,12)(10,11)(14,19)(15,18)(16,17)

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D7×Dic5, C3×D5×Dic7, C3×#63305, …
maximal normal subgroup ofC3×D7×Dic5, C3×D5×Dic7, C3×#63305, …
maximal quotient ofC3×C7⋊#168, C2×C3×Dic35, C9×Dic35, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 7);