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C3×(C3⋊D3)⋊C3×AGL(1,ℤ/9) #72254044

order8748
automorphism?
homologyAbelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C34

Aliases

direct productC3×(C3⋊D3)⋊C3×AGL(1,ℤ/9)

Splittings

Fitting, SylowC3×H(3,𝔽3)×3−1+2 ⋊ C22
derivedC32×C9 ⋊ C22×C33

Subgroups

subgroupG/H
center, hypercenterC3(C3⋊D3)⋊C3×AGL(1,ℤ/9)
commutator, cosocle, hypocenterC32×C9C22×C33
FrattiniC32C33×D32
FittingC3×H(3,𝔽3)×3−1+2C22
socleC33C32×D32
LCSG ▷ C32×C9
UCSC1 ◁ C3

4772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,3)(5,7)(6,9)(8,10)(11,12)(13,17,16,15,21,18,19,14,20);(4,5,7)(6,8,11)(9,12,10)(14,20,21,16,17,18)(15,19);(5,8,12)(7,10,11)(14,17,21)(16,20,18)

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of#72254055, D3×(C3⋊D3)⋊C3×AGL(1,ℤ/9)
maximal normal subgroup of#72254055, D3×(C3⋊D3)⋊C3×AGL(1,ℤ/9)
maximal quotient of#72254055

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3)(5,7)(6,9)(8,10)(11,12)(13,17,16,15,21,18,19,14,20), (4,5,7)(6,8,11)(9,12,10)(14,20,21,16,17,18)(15,19), (5,8,12)(7,10,11)(14,17,21)(16,20,18));