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Cayley diagram

C3×#1039 #64474

order300
isoclinismisoclinic class of C5⋊D5
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C5

Aliases

direct productC3×#1039

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊D5
commutator, hypocenterC52C3×C4
FrattiniC2C3×C5⋊D5
Fitting, socle, prime radicalC2×C3×C52C2
cosocleC2×C52C2×C3
LCSG ▷ C52
UCSC1 ◁ C2×C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 300

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11925399, #11925400, #11925401, …
maximal subgroup ofC3×#6033, C3×#6034, C3⋊#6033, …
maximal normal subgroup ofC3×#6033, C3×#6034, C3⋊#6033, …
maximal quotient ofC3×#6029, C2×C3×#1039, C9×#1039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 17);