Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#23472 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | C2×#393 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C23×C4×C79 ◁ G | |
1148 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)(3,8)(4,16,18,27)(5,10)(6,21)(7,11)(9,19,26,20)(12,25,28,32)(13,23)(14,24)(15,30)(17,22,29,31)(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)(2,8)(4,17,18,29)(5,14)(6,15)(7,13)(9,25,26,32)(10,24)(11,23)(12,20,28,19)(16,31,27,22)(21,30);(1,4)(2,9)(3,12)(5,20)(6,19)(7,18)(8,22)(10,27)(11,26)(13,28)(14,17)(15,29)(16,21)(23,31)(24,32)(25,30);(3,13)(8,23)(12,28)(14,15)(17,29)(22,31)(24,30)(25,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16460);