Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C22×D3×D42 (order 1536) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #343 |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(8,9,10,13,11,12,14,15);(2,3)(4,5,6,7)(9,12)(10,14)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,d^2=f,b^a=be,d^a=df,g^a=g^2,e^b=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #68735, #68736, #68763, … |
| maximal normal subgroup of | #68735, #68736, #68763, … |
| maximal quotient of | #67659, #67668, #67669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 113);