Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#3003 (order 768) · Inn(G) = D3×D4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #444 |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7)(5,12)(6,14)(8,18)(9,19)(10,11)(13,16)(15,17);(4,6)(5,11)(7,15)(8,12)(9,17)(10,16)(13,19)(14,18);(2,3)(4,5,8,13,10,15,9,14)(6,11,12,19,16,7,17,18)Polycyclic presentation:
2^6,3|a^2=d,d^2=f,e^2=f,c^a=cf,g^a=g^2,c^b=ce,e^b=ef,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69248, #69249, #71515, … |
| maximal normal subgroup of | #69248, #69249, #71515, … |
| maximal quotient of | #70288, #70290, #70291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 756);