Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×#67 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #477 |
96 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)(4,7,8,6)(5,9,10,11)(12,14)(13,17)(15,19)(16,18);(4,6,8,7)(5,9,10,11)(12,13,16,15)(14,17,18,19);(2,3)(4,5,8,10)(6,11,7,9)(13,15)(14,17)(18,19)Polycyclic presentation:
2^6,3|a^2=ef,b^2=e,c^2=ef,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #68246, #68247, #68253, … |
| maximal normal subgroup of | #68246, #68247, #68253, … |
| maximal quotient of | #69461, #69466, #69489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 580);