Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = D3×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#916 |
| Fitting | C2×C3×SD16 ⋊ C2 |
| Sylow | C3 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×D4 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C2×C3×SD16 | C2 |
| socle | C22×C3 | C2×D4 |
| hypercenter | C2×SD16 | D3 |
| hypocenter | C3 | C2×#135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×SD16 | |
266 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5)(6,8,10,16)(7,12,11,14)(9,18,17,15)(13,20,19,21);(1,2)(7,11)(8,15)(9,17)(12,20)(14,21)(16,18);(4,5)(6,7)(8,14)(9,13)(10,11)(12,16)(15,20)(17,19)(18,21);(1,2)42 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#916, #79650, #79674, … |
| maximal normal subgroup of | C4×#916, #79650, #79674, … |
| maximal quotient of | C2×#4756, C2×#4763, C2×#4779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1320);