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) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ Q8⋊C8 |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,6,7,11,8,5,9,10)(13,17,19,18);(2,3)(4,5,7,10,8,6,9,11)(12,13,15,18,16,19,14,17)Polycyclic presentation:
2^6,3|a^2=d,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×#204, C8×Dic12, #68055, … |
| maximal normal subgroup of | C8×#204, C8×Dic12, #68055, … |
| maximal quotient of | #67478, #67480, #67506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 15);