Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = C24×D3×D4 (order 768) · Inn(G) = D12 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C4×C8 ⋊ C2 |
| Sylow | C3 ⋊ D4⋊C8 |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,6,8,7,9,10,11)(13,17,19,18);(2,3)(12,13)(14,17)(15,18)(16,19)Polycyclic presentation:
2^6,3|b^2=cd,c^2=e,d^2=f,b^a=bc,c^a=ce,g^a=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D24, #68052, C8×#204, … |
| maximal normal subgroup of | C8×D24, #68052, C8×#204, … |
| maximal quotient of | #67441, #67443, #67477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 18);