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Cayley diagram

C3×#508 #5382

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) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#508

Subgroups

subgroupG/H
centerC3×C4D8
commutator, Frattini, cosocleC8C23×C3
socleC2×C3C2×D8
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

84 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67828, #67829, #71508, …
maximal subgroup of#67828, #67829, C3×#1417, …
maximal normal subgroup of#67828, #67829, C3×#1417, …
maximal quotient ofC3×#2138, C3×#2158, C3×C4×D16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 941);