Finite Group Databasebeta

← Prev (#4596) Next (#4598) →

Cayley diagram

C3×#350 #4597

order192
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C22×C4×D4 (order 128) · Inn(G) = D4 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC3×#350

Subgroups

subgroupG/H
centerC3×C8D4
commutatorC4C2×C3×C8
Frattini, cosocleC2×C8C22×C3
socleC2×C3C22⋊C8
prime radicalC3×#41C4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C8 ◁ C2×C3×C8 ◁ G

56 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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#69122, #69123, D3×#350, …
maximal subgroup of#69122, #69123, D3×#350, …
maximal normal subgroup of#69122, #69123, D3×#350, …
maximal quotient ofC3×#1315, C3×#1323, C3×D4⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 156);