Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×C4×D3×S4 (order 1152) · Inn(G) = C22×D3 · Out(G) = C2×C4×D3 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×D3 |
| commutator, socle | C2×C3 | C22×C8 |
| Fitting | C3×#131 | C2 |
| prime radical | C3×#41 | C4 |
| cosocle | C3×C8 | C23 |
| hypercenter | #131 | D3 |
| hypocenter | C3 | #504 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ #131 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(6,14)(10,20)(15,26)(16,27)(21,30)(22,31)(28,34)(32,35);(4,6)(5,10)(7,15)(8,16)(9,14)(11,21)(12,22)(13,20)(17,28)(18,26)(19,27)(23,32)(24,30)(25,31)(29,34)(33,35);(2,3)(4,5,7,11,8,12,17,23,9,13,18,24,19,25,29,33)(6,10,15,21,16,22,28,32,14,20,26,30,27,31,34,35)Polycyclic presentation:
2^6,3|a^2=d,d^2=e,e^2=f,g^a=g^2,c^b=cf60 conjugacy classes — view the full character table.
| maximal subgroup of | #69122, #69123, #70923, … |
| maximal normal subgroup of | #69122, #69123, #70923, … |
| maximal quotient of | #69881, #69882, #69883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 699);