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D3×AGL(1,𝔽5)×AGL(1,ℤ/8) #72229689

order3840
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productD3×AGL(1,𝔽5)×AGL(1,ℤ/8)

Splittings

FittingC3×C5×AGL(1,ℤ/8) ⋊ C2×C4
HallC3×C5 ⋊ C2×C4×AGL(1,ℤ/8)
SylowC5 ⋊ C4×D3×AGL(1,ℤ/8)
SylowC3 ⋊ C2×AGL(1,𝔽5)×AGL(1,ℤ/8)

Subgroups

subgroupG/H
centerC2C2×D3×D4×AGL(1,𝔽5)
commutatorC3×C4×C5C24×C4
FrattiniC4C23×D3×AGL(1,𝔽5)
FittingC3×C5×AGL(1,ℤ/8)C2×C4
socleC2×C3×C5C22×C4×D4
prime radicalD3×D5×AGL(1,ℤ/8)C2
cosocleC3×C4×D5C25
hypercenterAGL(1,ℤ/8)D3×AGL(1,𝔽5)
hypocenterC3×C5C2×C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

6484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 5 generators on 16 points:

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#10595100, D3×AGL(1,𝔽5)×#453, D3×#72213282
maximal normal subgroup ofD3×AGL(1,𝔽5)×#453
maximal quotient ofAGL(1,𝔽5)×#10595100, D3×#72213282
automorphism group ofD120, Dic60
holomorph ofC3×C5×C8

GAP

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

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