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D3×AGL(1,𝔽5)×#120 #72229520

order3840
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C211

Aliases

direct productD3×AGL(1,𝔽5)×#120

Splittings

FittingC3×C5×#120 ⋊ C2×C4
HallC3×C5 ⋊ C2×C4×#120
SylowC5 ⋊ C4×D3×#120
SylowC3 ⋊ C2×AGL(1,𝔽5)×#120

Subgroups

subgroupG/H
center, FrattiniC22C23×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C24×C4
FittingC3×C5×#120C2×C4
prime radicalD3×D5×#120C2
cosocleC22×C3×D5C25
hypercenter#120D3×AGL(1,𝔽5)
hypocenterC3×C5C2×C4×#120
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #120

17164 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)(14,16);(10,12)(13,14)(15,16);(2,3,4,5);(7,8);(9,10)(11,12)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽5)×B(4,𝔽2), D3×AGL(1,𝔽5)×#1020, C3⋊D3×AGL(1,𝔽5)×#120
maximal normal subgroup ofD3×AGL(1,𝔽5)×B(4,𝔽2)
maximal quotient ofD3×AGL(1,𝔽5)×B(4,𝔽2), D3×AGL(1,𝔽5)×#453, C3⋊D3×AGL(1,𝔽5)×#120
automorphism group ofC4×Dic15, C4⋊Dic15, #6719, …

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