Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #478 |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(13,17)(14,19)(16,18);(4,6,7,11)(5,8,9,10)(13,16)(17,18);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,16)(14,17,19,18)Polycyclic presentation:
2^6,3|a^2=e,b^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 | #79569, #79572, #79575, … |
| maximal normal subgroup of | #79569, #79572, #79575, … |
| maximal quotient of | #69459, #69461, #69468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 568);