Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×D12 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #428 |
118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(8,9,10,12,11,13,14,15);(9,13)(12,15);(2,3)(4,5,6,7)(9,12)(10,14)(13,15)Polycyclic presentation:
2^6,3|a^2=d,c^2=e,e^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #68735, #68736, #68758, … |
| maximal normal subgroup of | #68735, #68736, #68758, … |
| maximal quotient of | #69814, #69815, #69816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 680);