Finite Group Databasebeta

← Prev (#6305) Next (#6307) →

Cayley diagram

C2×C3×C3⋊Dic3 #6306

order216
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = C2×D4×AGL(2,𝔽3) (order 6912) · Inn(G) = C3⋊D3 · Out(G) = C2×D4×S4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C3⋊Dic3
unambiguous semidirect product(C2×C3)⋊(C3×Dic3)

Splittings

derivedC32 ⋊ C2×C3×C4
SylowC33 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22×C3C3⋊D3
commutator, hypocenterC32C2×C3×C4
FrattiniC2C2×C3×C3⋊D3
Fitting, socle, prime radicalC22×C33C2
cosocleC2×C32C22×C3
LCSG ▷ C32
UCSC1 ◁ C22×C3

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13154970, #13154971, #13154972, …
maximal subgroup ofC3×Dic32, C3×(C2×D3)⋊Dic3, C3×Dic3⋊Dic3, …
maximal normal subgroup ofC3×Dic32, C3×(C2×D3)⋊Dic3, C3×Dic3⋊Dic3, …
maximal quotient ofC2×C3×C3⋊(C3⋊C8), C3×#3748, C3×C4×C3⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 143);