Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C2×#6342371 (order 1024) · Inn(G) = C24 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#551 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3 |
| prime radical | C3×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
248 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,19);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16);(10,14)(13,16);(3,8)(5,7)75 conjugacy classes — view the full character table.
| Fitting subgroup of | #71922, #86790, #86791, … |
| maximal subgroup of | #71922, C3×#2197, #86790, … |
| maximal normal subgroup of | #71922, C3×#2197, #86790, … |
| maximal quotient of | C3×#2342, C3×#2343, C3×#2353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1440);