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Cayley diagram

C3×#368 #4615

order192
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C22×AGL(1,ℤ/16) (order 512) · Inn(G) = D8 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#368

Subgroups

subgroupG/H
centerC3×C4D8
commutatorC8C2×C3×C4
Frattini, cosocleC2×C8C22×C3
socleC2×C3#107
prime radicalC22×C3Q16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

40 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67581, #67582, #67583, …
maximal subgroup of#67581, #67582, #67583, …
maximal normal subgroup of#67581, #67582, #67583, …
maximal quotient ofC3×#1371, C3×#1372, C3×#1381, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 174);