Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#12670 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | D8 |
| commutator | C8 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C4×D8 |
| prime radical | C23×C79 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(5,6,8,13,10,15,22,30,11,16,23,31,25,33,9,14)(7,17,19,29,20,32,18,34,21,35,24,12,26,27,36,28)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,3,4)(5,7)(6,12)(8,18)(9,24)(10,26)(11,21)(13,17)(14,32)(15,34)(16,29)(19,22)(20,25)(23,36)(27,30)(28,33)(31,35);(5,8,10,22,11,23,25,9)(6,13,15,30,16,31,33,14)(7,18,26,19,21,36,20,24)(12,17,34,27,29,35,28,32)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5658);