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3−1+2×AGL(1,ℤ/16) #72244361

order3456
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22

Aliases

direct product3−1+2×AGL(1,ℤ/16)

Subgroups

subgroupG/H
center, socleC2×C3C32×#442
commutatorC3×C8C22×C32×C4
Frattini, cosocleC2×C3×C8C23×C32
prime radicalC32×#509C2×C3
LCSG ▷ C3×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C4×3−1+2 ◁ 3−1+2×#97 ◁ G

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

253 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofAGL(1,ℤ/9)×AGL(1,ℤ/16)
maximal subgroup ofAGL(1,ℤ/9)×AGL(1,ℤ/16)
maximal normal subgroup ofAGL(1,ℤ/9)×AGL(1,ℤ/16)

GAP

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

gap> G := Group((1,2,3,6,4,7,9,14,5,8,10,13,11,15,16,12)(18,20,23)(21,25,24), (2,6,8,13)(3,9)(4,11)(7,12,15,14)(10,16)(17,18,21,19,20,24,22,23,25), (2,7,8,15)(3,10)(6,12,13,14)(9,16));