Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = #67 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#950 |
| Fitting | C2×C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #67 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C2×C3×#41 | C2 |
| socle | C22×C3 | C2×D4 |
| hypercenter | C2×#41 | D3 |
| hypocenter | C3 | C2×#135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41 | |
266 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(3,4,5)(7,12)(9,17)(11,18)(13,14)(15,21)(19,20);(1,2)(6,8,10,16)(7,11,14,20)(9,15,17,21)(12,18,13,19);(4,5)(6,7)(8,11)(9,12)(10,14)(13,17)(15,18)(16,20)(19,21);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #69999, #70000, #70011, … |
| maximal normal subgroup of | #69999, #70000, #70011, … |
| maximal quotient of | C2×#4998, C2×#5002, C2×#5003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1380);