Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12957 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #442 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C2×#97×C79 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2)(3,4)(5,6,10,16,12,17,11,28)(7,18,21,29,23,33,22,36)(8,24,9,27,26,13,25,15)(14,19,30,20,31,35,32,34)(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);(2,4)(6,13,17,24)(8,25)(9,26)(10,11)(14,29,31,36)(15,16,27,28)(18,32,33,30)(19,34)(20,35)(21,22);(5,7,12,23)(6,14,17,31)(8,20,26,34)(9,19,25,35)(10,22,11,21)(13,29,24,36)(15,18,27,33)(16,32,28,30)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5945);