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) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #464 |
112 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5)(6,7)(9,14)(10,17)(11,19)(13,20)(15,16)(21,22);(4,5,6,7)(8,10)(9,13)(11,18)(12,17)(14,22)(15,20)(16,21)(19,23);(2,3)(4,5,6,7)(8,9,12,16)(10,13,17,21)(11,14,19,15)(18,22,23,20)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,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 | #80504, #80506, #80510, … |
| maximal normal subgroup of | #80504, #80506, #80510, … |
| maximal quotient of | #69540, #69541, #69559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 613);