Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×#67 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #428 |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(4,5,6,7)(9,13)(10,15)(12,14);(9,12)(13,14);(2,3)(8,9,10,14,11,12,15,13)Polycyclic presentation:
2^6,3|a^2=e,c^2=d,e^2=f,b^a=bf,c^a=ce,g^a=g^2,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #68980, #68981, #68982, … |
| maximal normal subgroup of | #68980, #68981, #68982, … |
| maximal quotient of | #69468, #69481, #69506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 523);