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Cayley diagram

C3×Q8×D5 #6800

order240
automorphismAut(G) = C22×AGL(1,𝔽5)×S4 (order 1920) · Inn(G) = C22×D5 · Out(G) = C23×D3
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×Q8×D5

Splittings

FittingC3×C5×Q8 ⋊ C2
HallC3×C5 ⋊ C2×Q8
SylowC5 ⋊ C2×C3×Q8

Subgroups

subgroupG/H
centerC2×C3C22×D5
commutator, cosocleC2×C5C23×C3
FrattiniC2C22×C3×D5
FittingC3×C5×Q8C2
socleC2×C3×C5C23
prime radicalC2×C3×D5C22
hypercenterC3×Q8D5
hypocenterC5C2×C3×Q8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×Q8

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#4009, #91238, D5×#215, …
maximal normal subgroup ofC3×#4009, #91238, D5×#215, …
maximal quotient ofC3×#4033, C3×#4034, C3×#4035, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 161);