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.
| 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)(6,10)(8,11)(13,18)(14,15)(17,19);(4,6)(5,8)(7,10)(9,11)(12,14,16,15)(13,17,19,18);(2,3)(4,5,7,9)(6,8,10,11)(12,13,16,19)(14,17,15,18)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #79992, #79993, #80010, … |
| maximal normal subgroup of | #79992, #79993, #80010, … |
| maximal quotient of | #69503, #69506, #69507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 594);