Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12567 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #102 |
| commutator | C8 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×#102 |
| prime radical | C24×C79 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
586 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)(7,18)(8,13)(9,25)(10,14)(11,17)(12,20)(15,23)(16,24)(19,34)(21,30)(22,33)(26,31)(27,32)(28,35)(29,36)(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);(5,7,10,21,8,19,23,20,11,22,27,35,24,36,9,26)(6,12,15,29,16,30,14,33,17,31,25,34,13,28,32,18);(5,8,11,24)(6,13,17,16)(7,19,22,36)(9,10,23,27)(12,28,31,30)(14,15,32,25)(18,34,33,29)(20,35,26,21)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5555);