Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33443 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | C22×D4 |
| commutator | C4 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×C8×D4 |
| prime radical | C2×#135×C79 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C2×M32×C79 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,6,13,7,12,24,30,8,11,25,29,5,21,23,15)(3,14,16,4,10,9,26,20,17,28,31,18,22,27,32,19)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3)(2,9)(4,15)(5,22)(6,16)(7,10)(8,17)(11,27)(12,28)(13,20)(14,21)(18,30)(19,29)(23,32)(24,26)(25,31);(1,4,6,9,7,20,24,28,8,18,25,27,5,19,23,14)(2,10,13,26,12,17,30,31,11,22,29,32,21,3,15,16);(2,11)(4,18)(9,27)(12,21)(13,29)(14,28)(15,30)(19,20)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26431);