Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#937 |
| Fitting | C22×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D3 |
| commutator, cosocle | C22×C3 | C24 |
| Frattini | C22 | C23×D3 |
| Fitting | C22×C3×D4 | C2 |
| socle | C23×C3 | C23 |
| hypercenter | C22×D4 | D3 |
| hypocenter | C3 | C2×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
662 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 143 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)(7,9)(11,13);(1,2)(6,7)(8,9);(4,5)(10,11)(12,13);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4471, #70022, #70036, … |
| maximal normal subgroup of | C2×#4471, #70022, #70036, … |
| maximal quotient of | C2×#4946, C2×#4947, C2×#4955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1358);