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 ⋊ #339 |
110 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)(8,10,12,14,11,15,13,9);(2,3)(4,5,6,7)(8,9,11,14)(10,13,15,12)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^2,e^b=ef48 conjugacy classes — view the full character table.
| maximal subgroup of | #68711, #68716, #68730, … |
| maximal normal subgroup of | #68711, #68716, #68730, … |
| maximal quotient of | #67659, #67665, #67667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 114);