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AGL(1,ℤ/8)×C3⋊S4 #72258732

order2304
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26×C3

Aliases

direct productAGL(1,ℤ/8)×C3⋊S4

Splittings

FittingC22×C3×AGL(1,ℤ/8) ⋊ D3

Subgroups

subgroupG/H
centerC2C2×D4×C3⋊S4
commutator, cosocleC3×C4×A4C24
FrattiniC4C23×C3⋊S4
FittingC22×C3×AGL(1,ℤ/8)D3
socleC23×C3C2×D3×D4
hypercenterAGL(1,ℤ/8)C3⋊S4
hypocenterC3×A4C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

3894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×S4×AGL(1,ℤ/8)
maximal normal subgroup ofD3×S4×AGL(1,ℤ/8)

GAP

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

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