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 ⋊ #443 |
106 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 106 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)(4,7,8,15,10,16,9,6)(5,11,12,18,13,19,17,14);(4,6)(5,11)(7,8)(9,16)(10,15)(12,18)(13,19)(14,17);(2,3)(4,5)(6,14)(7,11)(8,12)(9,17)(10,13)(15,18)(16,19)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=cef,e^b=ef,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4824, #78980, #78989, … |
| maximal normal subgroup of | C2×#4824, #78980, #78989, … |
| maximal quotient of | #68148, #68149, #68156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 383);