Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C2×#63011 (order 512) · Inn(G) = C2×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#421 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D4 |
| commutator | C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C4 | C23×C3 |
| socle | C2×C3 | C2×C22⋊C4 |
| prime radical | C2×C3×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
154 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 19 points:
(1,2,5,8,6,9,4,13)(3,10,12,7,11,16,14,15)(17,18,19);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13);(3,11)(7,15)(10,16)(12,14)66 conjugacy classes — view the full character table.
| Fitting subgroup of | #70394, #70395, #70396, … |
| maximal subgroup of | #70394, #70395, #70396, … |
| maximal normal subgroup of | #70394, #70395, #70396, … |
| maximal quotient of | C3×#1477, C3×#1478, C3×#1479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 854);