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Cayley diagram

C5×ΓL(1,𝔽25) #6689

order240
isoclinismisoclinic class of Dic6
automorphismAut(G) = C23×C4×D3 (order 192) · Inn(G) = C2×D3 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽25)

Splittings

FittingC3×C5×C8 ⋊ C2
HallC3×C5 ⋊ M16
SylowC3 ⋊ C5×M16

Subgroups

subgroupG/H
centerC4×C5C2×D3
commutatorC2×C3C2×C4×C5
FrattiniC4C2×C5×D3
FittingC3×C5×C8C2
socleC2×C3×C5C2×C4
prime radicalC2×C5×D3C4
cosocleC3×C4C22×C5
hypercenterC5×C8D3
hypocenterC3C5×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C5×C8

40 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)(4,8,6,11,7,5,10,9)(12,13,14,15,16);(2,3)(5,8)(9,11)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×ΓL(1,𝔽25), #91246, #91253, …
maximal normal subgroup ofD5×ΓL(1,𝔽25), #91246, #91253, …
maximal quotient ofC5×Dic3⋊C8, C5×#815, C5×(C2×D3)⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 50);