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) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #498 |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,11)(5,8,9,10)(13,16)(14,18)(17,19);(12,13,14,16,15,17,18,19);(2,3)(4,5,7,9)(6,10,11,8)Polycyclic presentation:
2^6,3|a^2=d,b^2=e,c^2=d,e^2=f,c^a=cd,g^a=g^2,c^b=cef,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69246, #69315, C2×#4855, … |
| maximal normal subgroup of | #69246, #69315, C2×#4855, … |
| maximal quotient of | #68803, #68827, #68835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 414);