Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#612424 (order 3072) · Inn(G) = C2×#67 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ #447 |
198 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 51 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);(9,12)(10,15)(13,14);(2,3)(4,5)(6,7)(8,9)(10,12)(11,14)(13,15)Polycyclic presentation:
2^6,3|e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^2,e^b=ef39 conjugacy classes — view the full character table.
| maximal subgroup of | #69696, #69697, #69702, … |
| maximal normal subgroup of | #69696, #69697, #69702, … |
| maximal quotient of | #71555, #71560, #71606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 776);