Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #419 |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,6,7,5)(8,11,12,19,13,20,10,17)(9,15,14,18,16,23,21,22);(8,10,13,12)(9,14,16,21)(11,19,20,17)(15,22,23,18);(2,3)(4,5,7,6)(8,9)(10,14)(11,18)(12,21)(13,16)(15,19)(17,23)(20,22)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,c^2=de,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #68712, #68725, #68759, … |
| maximal normal subgroup of | #68712, #68725, #68759, … |
| maximal quotient of | #69813, #69822, #69828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 679);