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Cayley diagram

C3×#622 #6280

order216
isoclinismisoclinic class of #157
automorphismAut(G) = C22×AΓL(1,𝔽9) (order 576) · Inn(G) = #157 · Out(G) = C24
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C3

Aliases

direct productC3×#622

Splittings

derivedC32 ⋊ C3×C8
SylowC33 ⋊ C8

Subgroups

subgroupG/H
center, hypercenterC2×C3#157
commutator, hypocenterC32C3×C8
FrattiniC2C3×#157
Fitting, socle, prime radicalC2×C33C4
cosocleC3⋊Dic3C2×C3
LCSG ▷ C32
UCSC1 ◁ C2×C3

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13165095, #13165096, #13165097, …
maximal subgroup ofC3×#3771, C3⋊#3771, D3×#622, …
maximal normal subgroup ofC3×#3771, C3⋊#3771, D3×#622, …
maximal quotient ofC3×#3708, C2×C3×#622, C9×#622, …
outer automorphism group of#6897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 117);