Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#12596 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #117 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | #432 |
| prime radical | C23×C79 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×#97×C79 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(5,6,10,16,11,15,25,14,12,17,8,23,9,27,26,33)(7,18,19,32,21,31,35,36,22,34,24,29,20,13,28,30)(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)(6,13,17,31)(8,24)(9,20)(10,19)(11,21)(12,22)(14,32,33,29)(15,18,27,34)(16,30,23,36)(25,35)(26,28);(5,8,9,25,12,10,11,26)(6,14,15,23,17,33,27,16)(7,19,20,28,22,24,21,35)(13,29,18,30,31,32,34,36)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5584);