Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#62817 (order 1536) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #342 |
142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 48 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(8,10)(9,12,13,14)(11,15);(2,3)(4,5,6,7)(8,9)(10,14)(11,13)(12,15)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,b^a=be,d^a=df,g^a=g^2,e^b=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊#2104, #68951, #68953, … |
| maximal normal subgroup of | C3⋊#2104, #68951, #68953, … |
| maximal quotient of | #67612, #67617, #67618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 86);