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D3×AGL(1,𝔽5)×B(4,𝔽2) #72230839

order7680
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C211

Aliases

direct productD3×AGL(1,𝔽5)×B(4,𝔽2)

Splittings

FittingC3×C5×B(4,𝔽2) ⋊ C2×C4
HallC3×C5 ⋊ C2×C4×B(4,𝔽2)
SylowC5 ⋊ C4×D3×B(4,𝔽2)
SylowC3 ⋊ C2×AGL(1,𝔽5)×B(4,𝔽2)

Subgroups

subgroupG/H
centerC2D3×AGL(1,𝔽5)×#120
commutatorC23×C3×C5C24×C4
FrattiniC23C23×D3×AGL(1,𝔽5)
FittingC3×C5×B(4,𝔽2)C2×C4
socleC2×C3×C5C2×C4×#120
prime radicalD3×D5×B(4,𝔽2)C2
cosocleC23×C3×D5C25
hypercenterB(4,𝔽2)D3×AGL(1,𝔽5)
hypocenterC3×C5C2×C4×B(4,𝔽2)
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C23 ◁ B(4,𝔽2)

At least 6348 conjugacy classes of subgroups; only the 1350 normal subgroups are shown.

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽5)×#5934
automorphism group ofC2×Dic30, C2×D60
holomorph ofC2×C3×C4×C5

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