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Cayley diagram

D3×Q32 #4917

order192
automorphismAut(G) = C2×D3×AGL(1,ℤ/16) (order 1536) · Inn(G) = D3×D8 · Out(G) = C22×C4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productD3×Q32

Splittings

FittingC3×Q32 ⋊ C2
SylowC3 ⋊ C2×Q32

Subgroups

subgroupG/H
centerC2D3×D8
commutator, cosocleC3×C8C23
FrattiniC8C22×D3
FittingC3×Q32C2
socleC2×C3C2×D8
prime radicalC2×D3D8
hypercenterQ32D3
hypocenterC3C2×Q32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

(1,3,6,14)(2,7,10,24)(4,16,19,32)(5,21,17,13)(8,26,27,31)(9,28,20,23)(11,29,30,15)(12,25,22,18)(33,34,35);(1,2,6,10)(3,11,14,30)(4,15,19,29)(5,20,17,9)(7,22,24,12)(8,25,27,18)(13,31,21,26)(16,28,32,23);(34,35)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×SD64, #69381, D3×Q64, …
maximal normal subgroup ofD3×SD64, #69381, D3×Q64, …
maximal quotient of#69207, #69211, #69215, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 476);