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Cayley diagram

C3×SD64 #4619

order192
nilpotency class5
isoclinismisoclinic class of D32
automorphismAut(G) = C2×#12067 (order 512) · Inn(G) = D16 · Out(G) = C2×C8
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×SD64

Subgroups

subgroupG/H
center, socleC2×C3D16
commutator, Frattini, cosocleC16C22×C3
prime radicalC3×D16C2
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×C16 ◁ G

32 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:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×SD64, #69379, #69380, …
maximal subgroup ofD3×SD64, #69379, #69380, …
maximal normal subgroup ofD3×SD64, #69379, #69380, …
maximal quotient ofC3×#1415, C3×#1416, C3×#1424, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 178);