Finite Group Databasebeta

← Prev (#5905) Next (#5907) →

Cayley diagram

C3×#576 #5906

order192
nilpotency class3
isoclinismisoclinic class of #576
automorphismAut(G) = C2×S4×AGL(1,ℤ/8) (order 1536) · Inn(G) = C22×D4 · Out(G) = C23×D3
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#576

Subgroups

subgroupG/H
center, socleC2×C3C22×D4
commutator, Frattini, cosocleC4C24×C3
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ G

268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,19);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16);(3,10)(7,14)(11,16)(13,15)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72122, #72123, #84007, …
maximal subgroup of#72122, #72123, C3×#2192, …
maximal normal subgroup of#72122, #72123, C3×#2192, …
maximal quotient ofC3×#2897, C3×#2913, C3×#2946, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1465);