Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#6628842 (order 3072) · Inn(G) = D3×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×#106 ⋊ C2 |
| Sylow | C3 ⋊ #427 |
114 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)(9,14)(10,16)(11,18)(13,19)(17,23)(20,21);(8,10,11,16,12,17,18,23)(9,13,15,20,14,21,22,19);(2,3)(8,9)(10,13)(11,15)(12,14)(16,20)(17,21)(18,22)(19,23)Polycyclic presentation:
2^6,3|b^2=e,c^2=d,e^2=f,c^a=cf,g^a=g^2,c^b=cef,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #69251, #69312, C2×#4860, … |
| maximal normal subgroup of | #69251, #69312, C2×#4860, … |
| maximal quotient of | #68527, #68530, #68533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 419);