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Cayley diagram

C3×C4⋊Dic5 #6681

order240
isoclinismisoclinic class of Dic10
automorphismAut(G) = C2×AGL(1,𝔽5)×#120 (order 1280) · Inn(G) = C2×D5 · Out(G) = C2×C4×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×C4⋊Dic5
unambiguous semidirect productC4⋊(C3×Dic5)

Splittings

HallC3×C5 ⋊ C4⋊C4
SylowC5 ⋊ C3×C4⋊C4

Subgroups

subgroupG/H
centerC22×C3C2×D5
commutatorC2×C5C2×C3×C4
FrattiniC22C2×C3×D5
FittingC2×C3×C4×C5C2
socle, prime radicalC22×C3×C5C22
cosocleC22×C5C22×C3
hypercenterC2×C3×C4D5
hypocenterC5C3×C4⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#90966, #90968, #90972, …
maximal normal subgroup of#90966, #90968, #90972, …
maximal quotient ofC3×C4⋊#168, C3×#3949, C3×#3950, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 42);