Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C210 (order 1024) · Inn(G) = C23 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#387 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C23 |
| commutator | C22 | C22×C3×C4 |
| Frattini, cosocle | C23 | C23×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
154 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,19);(1,3,4,2)(10,13)(14,16);(6,8)(9,11,12,15)(10,14,13,16)84 conjugacy classes — view the full character table.
| Fitting subgroup of | #74937, #74938, #74939, … |
| maximal subgroup of | #74937, #74938, #74939, … |
| maximal normal subgroup of | #74937, #74938, #74939, … |
| maximal quotient of | C3×#1434, C3×#1436, C3×#1441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 820);