Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = C2×C4×D3×D4 (order 384) · Inn(G) = D12 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C2×C3×C16 ⋊ C2 |
| 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)(4,6,7,16,9,18,22,32,10,19,23,33,8,17,21,31)(5,11,12,25,13,15,27,28,14,26,20,34,24,29,35,30);(2,3)(4,5)(6,15)(7,20)(8,24)(9,13)(10,14)(11,17)(12,23)(16,30)(18,26)(19,29)(21,27)(22,35)(25,32)(28,33)(31,34)Polycyclic presentation:
2^6,3|b^2=c,c^2=e,d^2=f,e^2=f,b^a=bd,c^a=cf,d^a=df,g^a=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | #69056, #69076, #69122, … |
| maximal normal subgroup of | #69056, #69076, #69122, … |
| maximal quotient of | #67503, #67506, #67523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 67);