Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C2×D3×#3484 (order 1536) · Inn(G) = #67 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #341 |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(8,10,11,19,13,20,12,21)(9,14,15,22,17,23,16,18);(2,3)(4,5,6,7)(8,9,12,16,13,17,11,15)(10,18,19,14,20,22,21,23)Polycyclic presentation:
2^6,3|a^2=d,b^2=c,c^2=f,d^2=f,b^a=bde,g^a=g^2,d^b=df48 conjugacy classes — view the full character table.
| maximal subgroup of | #68700, #68718, #68738, … |
| maximal normal subgroup of | #68700, #68718, #68738, … |
| maximal quotient of | #67662, #67667, #67669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 117);