Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C2×#72216616 (order 2048) · Inn(G) = C2×D4 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C3 |
| prime radical | C2×C3×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
258 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 162 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);(7,11)(8,14)(9,16)(12,19)(13,17)(18,20);(6,7,10,13)(8,12,14,20)(9,17,16,11)(15,19,21,18);(1,2)(6,8)(7,12)(9,15)(10,14)(11,18)(13,20)(16,21)(17,19)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #72100, #72101, #72102, … |
| maximal subgroup of | #72100, #72101, #72102, … |
| maximal normal subgroup of | #72100, #72101, #72102, … |
| maximal quotient of | C2×C3×#419, C2×C3×#417, C2×C3×#428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1463);