Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C29×D3 (order 3072) · Inn(G) = C22×D3 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #388 |
190 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(9,11)(13,15);(4,6,7,5)(12,13)(14,15);(2,3)(4,5,7,6)(8,9,10,11)Polycyclic presentation:
2^6,3|a^2=f,b^2=d,c^a=cdf,g^a=g^2,c^b=ce48 conjugacy classes — view the full character table.
| maximal subgroup of | #73440, C4×#879, #73448, … |
| maximal normal subgroup of | #73440, C4×#879, #73448, … |
| maximal quotient of | #69403, #69405, #69414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 504);