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Cayley diagram

C4×C52⋊C3 #64472

order300
isoclinismisoclinic class of C52⋊C3
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C5

Aliases

direct productC4×C52⋊C3
unambiguous semidirect product(C4×C52)⋊C3

Splittings

Fitting, HallC4×C52 ⋊ C3
derived, SylowC52 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC4C52⋊C3
commutator, hypocenterC52C3×C4
FrattiniC2C2×C52⋊C3
FittingC4×C52C3
socle, cosocleC2×C52C2×C3
prime radicalC2×C52⋊C3C2
LCSG ▷ C52
UCSC1 ◁ C4

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 60

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

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11925004, #11925005, #11925006, …
maximal subgroup of#10597064, #6029⋊C3, C8×C52⋊C3, …
maximal normal subgroup of#10597064, #6029⋊C3, C8×C52⋊C3, …
maximal quotient ofC8×C52⋊C3, C2×C4×C52⋊C3, C4×C52⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 15);