Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C27×D3 (order 768) · Inn(G) = C22×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C22⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #408 |
158 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 55 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)(8,10,11,16,12,17,18,22)(9,14,15,21,13,19,20,23);(4,5)(6,7)(9,13)(14,19)(15,20)(21,23);(2,3)(8,9)(10,14)(11,15)(12,13)(16,21)(17,19)(18,20)(22,23)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,b^a=bf,g^a=g^2,c^b=ce48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4727, #78600, #78604, … |
| maximal normal subgroup of | C2×#4727, #78600, #78604, … |
| maximal quotient of | #68104, #68105, #68122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 286);