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Cayley diagram

C2×D5×Dic3 #6778

order240
isoclinismisoclinic class of D3×D5
automorphismAut(G) = D3×AGL(1,𝔽5)×#120 (order 3840) · Inn(G) = D3×D5 · Out(G) = C2×#120
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×D5×Dic3

Splittings

derived, HallC3×C5 ⋊ C22×C4
SylowC5 ⋊ C22×Dic3
SylowC3 ⋊ C2×C4×D5

Subgroups

subgroupG/H
center, hypercenterC22D3×D5
commutator, hypocenterC3×C5C22×C4
FrattiniC2C2×D3×D5
Fitting, socleC22×C3×C5C22
prime radicalC22×C3×D5C2
cosocleC2×C3×C5C23
LCSG ▷ C3×C5
UCSC1 ◁ C22

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×AGL(1,𝔽5))⋊Dic3, C4×D5×Dic3, D5×Dic3⋊C4, …
maximal normal subgroup of(C2×AGL(1,𝔽5))⋊Dic3, C4×D5×Dic3, D5×Dic3⋊C4, …
maximal quotient ofC2×D5×C3⋊C8, D5×#208, #91283, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 139);