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Cayley diagram

C3×#1043 #64487

order300
isoclinismisoclinic class of #1043
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#1043

Splittings

Fitting, socle, HallC3×C52 ⋊ C4
derived, SylowC52 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC3#1043
commutator, hypocenterC52C3×C4
FrattiniC1G
Fitting, socleC3×C52C4
prime radicalC3×C5⋊D5C2
cosocleC5⋊D5C2×C3
LCSG ▷ C52
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 75

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#6053, C3⋊#6053, C3×D5⋊AGL(1,𝔽5), …
maximal normal subgroup ofC3×#6053, C3⋊#6053, C3×D5⋊AGL(1,𝔽5), …
maximal quotient ofC3×#6033, C2×C3×#1043, C9×#1043, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 30);