Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72214076 (order 6144) · Inn(G) = D3×D4 · Out(G) = #2403 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×D3×D8 |
| Fitting | C2×C3×D8 ⋊ C2 |
| Sylow | C3 ⋊ C22×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×D4 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C2×C3×D8 | C2 |
| socle | C22×C3 | C2×D4 |
| hypercenter | C2×D8 | D3 |
| hypocenter | C3 | C22×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
338 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 338 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) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(3,4,5)(6,7)(8,13)(9,11)(10,12);(1,2)(7,10)(8,12)(11,13);(4,5);(1,2)42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#357, #69190, #69191, … |
| maximal normal subgroup of | D3×#357, #69190, #69191, … |
| maximal quotient of | C2×#4756, C2×#4759, C2×#4760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1313);