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) = C4 |
No aliases for this group.
| Fitting | C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #482 |
106 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(12,14,16,18)(13,17,15,19);(5,8)(9,11)(12,13,16,15)(14,17,18,19);(2,3)(4,5)(6,9)(7,8)(10,11)(13,15)(14,17)(18,19)Polycyclic presentation:
2^6,3|b^2=f,c^2=e,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 | #78845, #78846, #78851, … |
| maximal normal subgroup of | #78845, #78846, #78851, … |
| maximal quotient of | #69460, #69461, #69462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 562);