Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8×C79 · Schur multiplier (H2) = C24×C4 |
| direct product | C79×#33326 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C24 |
| commutator | C2 | C22×C4×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C23×C8 |
| prime radical | C2×#41×C79 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ G | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 115 points:
(5,6,8,13,9,14,18,26,10,11,19,22,20,23,30,33)(7,12,16,24,17,25,29,34,15,21,27,31,28,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,3,4)(6,11)(12,21)(13,22)(14,23)(24,31)(25,32)(26,33)(34,36);(5,7)(6,12)(8,16)(9,17)(10,15)(11,21)(13,24)(14,25)(18,29)(19,27)(20,28)(22,31)(23,32)(26,34)(30,35)(33,36);(7,15)(12,21)(16,27)(17,28)(24,31)(25,32)(29,35)(34,36)10744 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26314);