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Cayley diagram

C3×C5×ΓL(1,𝔽8) #64651

order315
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C4×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C4×D3
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×ΓL(1,𝔽8)
unambiguous semidirect productC7⋊(C32×C5)
unambiguous semidirect product(C3×C7)⋊(C3×C5)
unambiguous semidirect product(C5×C7)⋊C32
unambiguous semidirect product(C3×C5×C7)⋊C3

Splittings

Fitting, socleC3×C5×C7 ⋊ C3
derived, SylowC7 ⋊ C32×C5
HallC5×C7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3×C5ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C32×C5
FrattiniC1G
Fitting, socleC3×C5×C7C3
LCSG ▷ C7
UCSC1 ◁ C3×C5

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 105

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C5×AGL(1,𝔽7), C3×D5×ΓL(1,𝔽8), C3×C5⋊AGL(1,𝔽7), …
maximal normal subgroup ofC3×C5×AGL(1,𝔽7), C3×D5×ΓL(1,𝔽8), C3×C5⋊AGL(1,𝔽7), …
maximal quotient ofC2×C3×C5×ΓL(1,𝔽8), C5×C9×ΓL(1,𝔽8), C5×#4427, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(315, 3);