Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72210469 (order 6144) · Inn(G) = C2×D12 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #487 |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9)(8,10)(12,13,14,16,15,17,18,19);(4,6,7,11)(5,8,9,10);(2,3)(4,5,7,9)(6,10,11,8)(13,16)(14,18)(17,19)Polycyclic presentation:
2^6,3|a^2=e,b^2=e,c^2=de,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^254 conjugacy classes — view the full character table.
| maximal subgroup of | #68059, #68061, #68063, … |
| maximal normal subgroup of | #68059, #68061, #68063, … |
| maximal quotient of | #68631, #68636, #68643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 240);