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S4×#949 #72274507

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

Aliases

direct productS4×#949

Splittings

FittingC22×C3×#41 ⋊ C2×D3

Subgroups

subgroupG/H
centerC2C2×#67×S4
commutator, cosocleC3×C4×A4C24
FrattiniC4C22×D3×S4
FittingC22×C3×#41C2×D3
socleC23×C3C2×D3×D4
hypercenter#41D3×S4
hypocenterC3×A4C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

3366 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:

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

Conjugacy classes & character table

105 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((1,2,4)(5,6,7)(9,13)(12,14), (6,7)(9,12)(10,15)(13,14), (2,3), (2,3)(8,9)(10,12)(11,13)(14,15));