Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#511790 (order 3072) · Inn(G) = #217 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #342 |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(4,5,6,7)(8,10)(9,12)(11,15)(13,14);(2,3)(8,9)(10,14,15,12)(11,13)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,b^a=be,e^a=ef,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #67448, #67449, #67450, … |
| maximal normal subgroup of | #67448, #67449, #67450, … |
| maximal quotient of | #67613, #67616, C3⋊#1304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 85);