Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = C2×D12 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C22⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #449 |
162 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 162 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) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(8,11,10,9,13,15,12,14);(4,5)(6,7)(8,10,13,12)(9,11,14,15);(2,3)(8,9)(10,11)(12,15)(13,14)Polycyclic presentation:
2^6,3|b^2=f,c^2=df,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce39 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4734, #78610, #78613, … |
| maximal normal subgroup of | C2×#4734, #78610, #78613, … |
| maximal quotient of | #68107, #68111, #68130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 293);