Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#935 |
| Sylow | C3 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D3 |
| commutator, cosocle | C22×C3 | C24 |
| Frattini | C22 | C23×D3 |
| Fitting, prime radical | C22×C3×D4 | C2 |
| socle | C23×C3 | C23 |
| hypercenter | C22×D4 | D3 |
| hypocenter | C3 | C2×#123 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 17 points:
(3,4,5)(6,7)(8,9)(10,12)(11,14)(13,16)(15,17);(1,2)(12,16)(14,17);(4,5)(7,9)(10,11,13,15)(12,14,16,17);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4472, #70023, #70029, … |
| maximal normal subgroup of | C2×#4472, #70023, #70029, … |
| maximal quotient of | C2×#4943, C2×#4946, C2×#4944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1355);