Finite Group DatabasebetaFinite Group Databasebeta| order | 3904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×D4 |
| commutator | C4 | C22×C4×C61 |
| Frattini, cosocle | C2×C4 | C23×C61 |
| prime radical | C2×D4×C61 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C42×C61 ◁ G | |
120 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)(62,63,65,64)(66,67)(68,70)(69,71)(72,73);(67,70)(68,73)(71,72);(62,64,65,63)(67,71)(70,72)1342 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3904, 121);