Finite Group Databasebeta

← Prev (#6245) Next (#6247) →

Cayley diagram

C3×C3⋊(C3⋊C8) #6246

order216
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) = C23×AGL(2,𝔽3) (order 3456) · Inn(G) = C3⋊D3 · Out(G) = C23×S4
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C3

Aliases

direct productC3×C3⋊(C3⋊C8)
unambiguous semidirect productC3⋊(C3×C3⋊C8)

Splittings

derivedC32 ⋊ C3×C8
SylowC33 ⋊ C8

Subgroups

subgroupG/H
center, hypercenterC3×C4C3⋊D3
commutator, hypocenterC32C3×C8
FrattiniC4C3×C3⋊D3
FittingC33×C4C2
socle, prime radicalC2×C33C4
cosocleC32×C4C2×C3
LCSG ▷ C32
UCSC1 ◁ C3×C4

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

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

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11722765, #11722766, #11722767, …
maximal subgroup ofC3×#3708, C3⋊#3708, C3×D3×C3⋊C8, …
maximal normal subgroup ofC3×#3708, C3⋊#3708, C3×D3×C3⋊C8, …
maximal quotient ofC3×C3⋊(C3⋊C16), C2×C3×C3⋊(C3⋊C8), C3×C3⋊(C9⋊C8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 83);