Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C22×D3×#3003 (order 3072) · Inn(G) = D3×D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ #467 |
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)(5,7)(8,10,12,18)(9,13,15,21)(11,19,20,17)(14,22,23,16);(8,9,11,14,12,15,20,23)(10,16,17,21,18,22,19,13);(2,3)(4,5)(6,7)Polycyclic presentation:
2^6,3|b^2=e,c^2=f,e^2=f,c^a=cd,g^a=g^2,c^b=ce,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69214, #69227, #69228, … |
| maximal normal subgroup of | #69214, #69227, #69228, … |
| maximal quotient of | #70244, #70248, #70254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 747);