Finite Group DatabasebetaFinite Group Databasebeta| order | 3904 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | D4 |
| commutator | C4 | C42×C61 |
| Frattini, cosocle | C22×C4 | C22×C61 |
| socle | C22×C61 | C4⋊C4 |
| prime radical | C23×C61 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C4×C61 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 2 generators on 81 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)(64,65)(66,67,69,74,71,75,70,73)(68,76,77,72,79,80,78,81);(63,65)(66,68,70,78,71,79,69,77)(67,72,74,80,75,81,73,76)1708 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3904, 22);