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 ⋊ #450 |
188 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(8,9,11,13)(10,15,14,12);(9,12)(10,14)(13,15);(2,3)(4,5)(6,7)Polycyclic presentation:
2^6,3|c^2=f,e^2=f,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 | #79651, #79654, #79659, … |
| maximal normal subgroup of | #79651, #79654, #79659, … |
| maximal quotient of | #70169, #70198, #70207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 728);