Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #441 |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7)(10,17)(13,22)(16,23)(18,20);(8,10,12,17)(9,13,15,22)(11,20,21,18)(14,23,19,16);(2,3)(8,9,12,15)(10,16,17,23)(11,19,21,14)(13,20,22,18)Polycyclic presentation:
2^6,3|a^2=f,b^2=f,c^2=d,e^2=f,b^a=be,e^a=ef,g^a=g^2,c^b=cf,e^b=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #79648, #79650, #79657, … |
| maximal normal subgroup of | #79648, #79650, #79657, … |
| maximal quotient of | #69458, #69464, #69475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 589);