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.
| Fitting | C3×C4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #464 |
112 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 43 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)(8,10,12,18)(9,14,13,22)(11,20,21,17)(15,23,19,16);(4,5,6,7)(9,13)(14,22)(15,19)(16,23);(2,3)(5,7)(8,9)(10,16)(11,19)(12,13)(14,17)(15,21)(18,23)(20,22)Polycyclic presentation:
2^6,3|b^2=ef,c^2=f,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 | #79624, #79627, #79639, … |
| maximal normal subgroup of | #79624, #79627, #79639, … |
| maximal quotient of | #69462, #69464, #69468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 587);