Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 3072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D3×C22⋊Q8 |
| Fitting | C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ C2×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×C22⋊Q8 | C2 |
| prime radical | C23×D3 | C22 |
| hypercenter | C22⋊Q8 | D3 |
| hypocenter | C3 | C2×C22⋊Q8 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
322 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 121 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 — 4 generators on 15 points:
(1,2,3)(5,7);(8,10,11,15)(9,12,13,14);(4,5)(6,7)(8,9,11,13)(10,14,15,12);(2,3)42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#328, #68335, #68336, … |
| maximal normal subgroup of | D3×#328, #68335, #68336, … |
| maximal quotient of | #73866, #73888, #73896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1185);