Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×D12 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #477 |
96 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)(6,10)(12,15,16,13,17,18,14,19);(4,6,7,10)(5,8,9,11)(12,14,17,16)(13,18,19,15);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,17,19)(14,18,16,15)Polycyclic presentation:
2^6,3|a^2=f,b^2=e,c^2=def,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce39 conjugacy classes — view the full character table.
| maximal subgroup of | #68188, #68189, #68202, … |
| maximal normal subgroup of | #68188, #68189, #68202, … |
| maximal quotient of | #68624, #68632, #68643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 409);