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Cayley diagram

D5×C3⋊C8 #6646

order240
isoclinismisoclinic class of D3×D5
automorphismAut(G) = C23×D3×AGL(1,𝔽5) (order 960) · Inn(G) = D3×D5 · Out(G) = C24
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

direct productD5×C3⋊C8

Splittings

derived, HallC3×C5 ⋊ C2×C8
SylowC5 ⋊ C2×C3⋊C8
SylowC3 ⋊ C8×D5

Subgroups

subgroupG/H
center, Frattini, hypercenterC4D3×D5
commutator, hypocenterC3×C5C2×C8
Fitting, cosocleC3×C4×C5C22
socleC2×C3×C5C2×C4
prime radicalC2×C3×D5C4
LCSG ▷ C3×C5
UCSC1 ◁ C4

44 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C3⋊C8, ΓL(1,𝔽16)⋊C8, #91149, …
maximal normal subgroup ofAGL(1,𝔽5)×C3⋊C8, ΓL(1,𝔽16)⋊C8, #91149, …
maximal quotient ofD5×C3⋊C16, #90934, Dic5×C3⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 7);