Finite Group Databasebeta

← Prev (#1105) Next (#1107) →

Cayley diagram

C3×C3⋊Dic3 #1106

order108
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = C22×AGL(2,𝔽3) (order 1728) · Inn(G) = C3⋊D3 · Out(G) = C22×S4
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×C3⋊Dic3
unambiguous semidirect productC3⋊(C3×Dic3)

Splittings

derivedC32 ⋊ C3×C4
SylowC33 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C3C3⋊D3
commutator, hypocenterC32C3×C4
FrattiniC2C3×C3⋊D3
Fitting, socle, prime radicalC2×C33C2
cosocleC2×C32C2×C3
LCSG ▷ C32
UCSC1 ◁ C2×C3

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

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

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

derived subgroup of#88935, #88936, #88937, …
maximal subgroup ofC3×#622, C3⋊#622, C3×D3×Dic3, …
maximal normal subgroup ofC3×#622, C3⋊#622, C3×D3×Dic3, …
maximal quotient ofC3×C3⋊(C3⋊C8), C2×C3×C3⋊Dic3, C3×C3⋊Dic9, …
outer automorphism group of#6860

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 33);