Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#63011 (order 1536) · Inn(G) = C2×D12 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #421 |
154 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,10,11,12)(6,13)(9,16,17,18)(14,19);(4,6)(5,9)(7,14)(8,13)(10,18)(11,17)(12,16)(15,19);(2,3)(4,5)(6,9)(7,10)(8,11)(12,15)(13,17)(14,18)(16,19)Polycyclic presentation:
2^6,3|c^2=d,e^2=f,c^a=ce,d^a=df,e^a=ef,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #68700, #68703, #68704, … |
| maximal normal subgroup of | #68700, #68703, #68704, … |
| maximal quotient of | #69430, #69440, #69441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 564);