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AGL(1,𝔽5)×S4×AGL(1,𝔽7) #72243849

order20160
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productAGL(1,𝔽5)×S4×AGL(1,𝔽7)

Splittings

Fitting, socleC22×C5×C7 ⋊ C2×C3×C4×D3
derivedC5×C7×A4 ⋊ C22×C3×C4
HallC5×C7 ⋊ C2×C3×C4×S4
SylowC7 ⋊ C2×C3×AGL(1,𝔽5)×S4
SylowC5 ⋊ C4×S4×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenterC5×C7×A4C22×C3×C4
Fitting, socleC22×C5×C7C2×C3×C4×D3
prime radicalD5×S4×AGL(1,𝔽7)C2
cosocleC7×D5×A4C23×C3
LCSG ▷ C5×C7×A4
UCSC1

1928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 140

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

holomorph ofC22×C5×C7

GAP

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

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