Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4513 |
| automorphism | Aut(G) = D3×#3109 (order 768) · Inn(G) = #817 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ #365 |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,6,9,7,11,12,16,8,10,13,15,14,17,19,18);(2,3)(5,9,10,15)(6,12)(7,14)(11,18,17,16)(13,19)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,d^2=e,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,c^b=cf36 conjugacy classes — view the full character table.
| maximal subgroup of | #69090, #69091, #69116, … |
| maximal normal subgroup of | #69090, #69091, #69116, … |
| maximal quotient of | #67513, #67816, C2×#4513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 72);