Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #826 |
| automorphism | Aut(G) = C2×D3×#442 (order 768) · Inn(G) = #213 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ #368 |
40 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)(4,6,7,15,9,18,20,17)(5,10,11,23,13,25,27,24)(8,21,19,16,22,32,34,33)(12,28,26,14,29,30,35,31);(2,3)(4,5,8,12,20,27,34,35,9,13,22,29,7,11,19,26)(6,14,16,23,15,30,32,25,18,31,33,24,17,28,21,10)Polycyclic presentation:
2^6,3|a^2=d,b^2=c,c^2=f,d^2=ef,e^2=f,b^a=bd,g^a=g^2,d^b=de,e^b=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69238, #69239, #69260, … |
| maximal normal subgroup of | #69238, #69239, #69260, … |
| maximal quotient of | #67469, #67470, #67728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 52);