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Cayley diagram

C3×D5⋊C8 #6750

order240
isoclinismisoclinic class of AGL(1,𝔽5)
automorphismAut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = AGL(1,𝔽5) · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC3×D5⋊C8
unambiguous semidirect productD5⋊(C3×C8)

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC3×C4AGL(1,𝔽5)
commutator, hypocenterC5C2×C3×C8
FrattiniC2C2×C3×AGL(1,𝔽5)
FittingC3×C4×C5C4
socleC2×C3×C5C2×C4
prime radicalC2×C3×D5C4
cosocleDic5C22×C3
LCSG ▷ C5
UCSC1 ◁ C3×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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽16)⋊C8, #91150, #91152, …
maximal normal subgroup ofΓL(1,𝔽16)⋊C8, #91150, #91152, …
maximal quotient ofC3×D5⋊C16, C3×#3990, C3×C4×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 111);