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Cayley diagram

C2×#737 #4214

order162
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphismAut(G) = C3×AGL(2,𝔽3) (order 1296) · Inn(G) = C32 · Out(G) = C3×GL(2,𝔽3)
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C32

Aliases

direct productC2×#737

Subgroups

subgroupG/H
centerC2×C9C32
commutator, Frattini, cosocleC3C2×C33
socleC2×C3C33
prime radicalC2×H(3,𝔽3)C3
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C2×C9 ◁ G

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2)(4,7,13)(8,16,14)(9,15,21)(17,24,22)(18,23,27)(25,29,28);(3,4,8)(5,9,17)(6,7,14)(10,18,25)(11,15,22)(12,13,16)(19,23,28)(20,21,24)(26,27,29);(3,5,10,6,11,19,12,20,26)(4,9,18,7,15,23,13,21,27)(8,17,25,14,22,28,16,24,29)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10619905, #10619907, #10619909, …
Fitting subgroup of#66409, #66410, C2×#4207, …
maximal subgroup of#66409, #66410, C4×#737, …
maximal normal subgroup of#66409, #66410, C4×#737, …
maximal quotient ofC4×#737, C22×#737, C2×#6887, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 50);