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D3×H(3,𝔽3)×AGL(1,ℤ/9) #72244058

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

Aliases

direct productD3×H(3,𝔽3)×AGL(1,ℤ/9)

Splittings

Fitting, SylowC3×H(3,𝔽3)×3−1+2 ⋊ C22

Subgroups

subgroupG/H
centerC3C32×D3×AGL(1,ℤ/9)
commutator, cosocleC32×C9C22×C33
FrattiniC32C33×D32
FittingC3×H(3,𝔽3)×3−1+2C22
socleC33C2×C33×D3
hypercenterH(3,𝔽3)D3×AGL(1,ℤ/9)
hypocenterC3×C9C22×C3×H(3,𝔽3)
LCSG ▷ C32×C9 ▷ C3×C9
UCSC1 ◁ C3 ◁ H(3,𝔽3)

5207 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of#72244059, D3×(C3⋊D3)⋊C3×AGL(1,ℤ/9)
maximal normal subgroup ofD3×(C3⋊D3)⋊C3×AGL(1,ℤ/9)
maximal quotient of#72244059

GAP

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

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