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Cayley diagram

C3×Q32 #856

order96
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C2×AGL(1,ℤ/16) (order 256) · Inn(G) = D8 · Out(G) = C22×C4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×Q32

Subgroups

subgroupG/H
center, socle, prime radicalC2×C3D8
commutator, Frattini, cosocleC8C22×C3
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 35 points:

(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)(33,34,35);(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)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#4521, #4522, D3×Q32, …
maximal subgroup of#4521, #4522, C3×SD64, …
maximal normal subgroup of#4521, #4522, C3×SD64, …
maximal quotient ofC3×#358, C3×#366, C2×C3×Q32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 63);