Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×D3×D4 (order 384) · Inn(G) = D3×D4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#104 ⋊ C2 |
| Sylow | C3 ⋊ #444 |
104 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 45 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)(4,7,8,17,9,16,18,6)(5,11,12,19,14,10,13,15);(4,6)(5,10)(7,8)(9,17)(11,14)(12,15)(13,19)(16,18);(2,3)(4,5)(6,15)(7,11)(8,12)(9,14)(10,16)(13,18)(17,19)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^2,c^b=ce,e^b=ef,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4823, #78980, #78989, … |
| maximal normal subgroup of | C2×#4823, #78980, #78989, … |
| maximal quotient of | #68150, #68151, #68152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 382);