Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C3×SD16 ⋊ C2 |
| Sylow | C3 ⋊ #452 |
152 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 152 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)(5,7)(8,10,12,17)(9,14,13,20)(11,19,18,16)(15,22,21,23);(9,13)(10,16)(11,18)(14,22)(17,19)(20,23);(2,3)(4,5)(6,7)(8,9)(10,14)(11,15)(12,13)(16,23)(17,20)(18,21)(19,22)Polycyclic presentation:
2^6,3|c^2=f,e^2=f,b^a=bf,c^a=cd,g^a=g^2,c^b=ce,e^b=ef,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #79652, #79655, #79657, … |
| maximal normal subgroup of | #79652, #79655, #79657, … |
| maximal quotient of | #70167, #70199, #70206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 732);