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Cayley diagram

C3×Q64 #4620

order192
nilpotency class5
isoclinismisoclinic class of D32
automorphismAut(G) = C2×AGL(1,ℤ/32) (order 1024) · Inn(G) = D16 · Out(G) = C22×C8
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×Q64

Subgroups

subgroupG/H
center, socle, prime radicalC2×C3D16
commutator, Frattini, cosocleC16C22×C3
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×C16 ◁ G

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 2 generators on 67 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)(36,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37);(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(17,49,33,65)(18,50,34,66)(19,51,35,67)

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67609, #67610, D3×Q64, …
maximal subgroup of#67609, #67610, C3×SD128, …
maximal normal subgroup of#67609, #67610, C3×SD128, …
maximal quotient ofC3×#1416, C3×#1423, C2×C3×Q64, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 179);