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C3×AGL(1,𝔽5)×#214 #72229587

order2880
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×AGL(1,𝔽5)×#214

Splittings

HallC32×C5 ⋊ C4×SD16
SylowC32 ⋊ SD16×AGL(1,𝔽5)
SylowC5 ⋊ C3×C4×#214

Subgroups

subgroupG/H
centerC2×C3AGL(1,𝔽5)×#67
commutatorC3×C4×C5C22×C3×C4
FrattiniC4C2×C3×D3×AGL(1,𝔽5)
FittingC32×C5×D4C2×C4
socleC2×C32×C5C4×D4
prime radicalC32×D4×D5C22
cosocleC3×C4×D5C23×C3
hypercenterC3×D4D3×AGL(1,𝔽5)
hypocenterC3×C5C3×C4×SD16
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#63930, C3×AGL(1,𝔽5)×#932, AGL(1,𝔽5)×#63937
maximal normal subgroup ofAGL(1,𝔽5)×#63930, C3×AGL(1,𝔽5)×#932, AGL(1,𝔽5)×#63937

GAP

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

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