Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4451 |
| automorphism | Aut(G) = C22×D3×#120 (order 768) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ #330 |
60 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)(5,8,9,10)(12,14,15,13,17,18,16,19);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,17,19)(14,15,18,16)Polycyclic presentation:
2^6,3|a^2=f,b^2=c,c^2=f,d^2=ef,b^a=bd,c^a=ce,d^a=def,g^a=g^2,d^b=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #68341, #68344, #68346, … |
| maximal normal subgroup of | #68341, #68344, #68346, … |
| maximal quotient of | #67477, #67478, #67659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 24);