Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = D12 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C3×C8 ⋊ C2 |
| Sylow | C3 ⋊ #416 |
158 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)(4,5,6,7)(8,11,10,17,13,20,12,19)(9,15,21,23,16,22,14,18);(8,10,13,12)(9,14,16,21)(11,17,20,19)(15,18,22,23);(2,3)(8,9)(10,14)(11,18)(12,21)(13,16)(15,19)(17,22)(20,23)Polycyclic presentation:
2^6,3|b^2=f,c^2=de,d^2=f,c^a=cd,d^a=df,g^a=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | #68646, #68694, #68918, … |
| maximal normal subgroup of | #68646, #68694, #68918, … |
| maximal quotient of | #69734, #69736, C4×#821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 672);