Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C2×C4×D3×D4 (order 384) · Inn(G) = #67 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #350 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,6)(5,11)(7,16)(8,23)(9,17)(10,18)(12,25)(13,26)(14,27)(15,22)(19,33)(20,24)(21,31)(28,34)(29,35)(30,32);(2,3)(4,5,7,12,9,13,20,28,10,14,21,29,8,22,19,32)(6,15,16,30,17,11,24,25,18,26,31,34,23,27,33,35)Polycyclic presentation:
2^6,3|a^2=c,c^2=e,d^2=f,e^2=f,b^a=bd,d^a=df,g^a=g^2,d^b=df48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#350, #69125, #70923, … |
| maximal normal subgroup of | D3×#350, #69125, #70923, … |
| maximal quotient of | #67671, #67689, #67718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 120);