Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = C2×#67 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#124 ⋊ C2 |
| Sylow | C3 ⋊ #495 |
130 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(9,14)(10,18)(11,20)(13,21)(15,16)(17,22);(4,5,6,7)(8,10)(9,13)(11,19)(12,18)(14,22)(15,21)(16,17)(20,23);(2,3)(5,7)(8,9)(10,17)(11,14)(12,16)(13,18)(15,20)(19,21)(22,23)Polycyclic presentation:
2^6,3|b^2=ef,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #80289, #80296, #80524, … |
| maximal normal subgroup of | #80289, #80296, #80524, … |
| maximal quotient of | #69549, #69551, #69558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 618);