Finite Group Databasebeta

← Prev (#6697) Next (#6699) →

Cayley diagram

C5×(C2×D3)⋊C4 #6698

order240
isoclinismisoclinic class of Dic6
automorphismAut(G) = C24×C4×D3 (order 384) · Inn(G) = C2×D3 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×(C2×D3)⋊C4
unambiguous semidirect product(C2×D3)⋊(C4×C5)

Splittings

FittingC2×C3×C4×C5 ⋊ C2
HallC3×C5 ⋊ C22⋊C4
SylowC3 ⋊ C5×C22⋊C4

Subgroups

subgroupG/H
centerC22×C5C2×D3
commutatorC2×C3C2×C4×C5
FrattiniC22C2×C5×D3
FittingC2×C3×C4×C5C2
socleC22×C3×C5C22
prime radicalC22×C5×D3C2
cosocleC22×C3C22×C5
hypercenterC2×C4×C5D3
hypocenterC3C5×C22⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C2×C4×C5

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 120

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

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#91326, #91341, #91342, …
maximal normal subgroup of#91326, #91341, #91342, …
maximal quotient ofC5×#805, C5×#806, C5×#809, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 59);