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Cayley diagram

C3×D5×C11 #66512

order330
isoclinismisoclinic class of D5
automorphismAut(G) = C22×C5×AGL(1,𝔽5) (order 400) · Inn(G) = D5 · Out(G) = C23×C5
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC3×D5×C11
unambiguous semidirect productC5⋊(C2×C3×C11)

Splittings

Fitting, socle, HallC3×C5×C11 ⋊ C2
derived, SylowC5 ⋊ C2×C3×C11
HallC5×C11 ⋊ C2×C3
HallC3×C5 ⋊ C2×C11

Subgroups

subgroupG/H
center, hypercenterC3×C11D5
commutator, cosocle, hypocenterC5C2×C3×C11
FrattiniC1G
Fitting, socleC3×C5×C11C2
LCSG ▷ C5
UCSC1 ◁ C3×C11

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 165

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3)(5,6)(7,8)(9,10,11,12,13,14,15,16,17,18,19);(4,5,7,8,6)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C11×AGL(1,𝔽5), C3×C11⋊AGL(1,𝔽5), C11×ΓL(1,𝔽16), …
maximal normal subgroup ofC3×C11×AGL(1,𝔽5), C3×C11⋊AGL(1,𝔽5), C11×ΓL(1,𝔽16), …
maximal quotient ofC3×C11×Dic5, C2×C3×D5×C11, C9×D5×C11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(330, 6);