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Cayley diagram

C3×C3⋊D21 #67404

order378
isoclinismisoclinic class of C3⋊D21
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

direct productC3×C3⋊D21
unambiguous semidirect productC3⋊(C3×D21)

Splittings

Fitting, socle, HallC33×C7 ⋊ C2
derivedC32×C7 ⋊ C2×C3
SylowC33 ⋊ D7
SylowC7 ⋊ C3×C3⋊D3

Subgroups

subgroupG/H
center, hypercenterC3C3⋊D21
commutator, cosocle, hypocenterC32×C7C2×C3
FrattiniC1G
Fitting, socleC33×C7C2
LCSG ▷ C32×C7
UCSC1 ◁ C3

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 126

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D7×C3⋊D3, C3×D3×D21, D3×C3⋊D21, …
maximal normal subgroup ofC3×D7×C3⋊D3, C3×D3×D21, D3×C3⋊D21, …
maximal quotient ofC3×C3⋊Dic21, C2×C3×C3⋊D21, C3×C3⋊D63, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 57);