Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12569 |
| 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 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
570 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)(7,16)(8,12)(9,23)(10,15)(11,18)(13,21)(14,22)(17,31)(19,28)(20,26)(24,29)(25,33)(27,34)(30,36)(32,35)(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);(5,7,9,19,22,35,36,33,10,20,24,18,8,17,21,34)(6,11,13,26,12,25,29,32,15,28,30,16,14,27,23,31);(5,8,10,22)(6,12,15,14)(7,17,20,35)(9,21,24,36)(11,25,28,27)(13,29,30,23)(16,31,26,32)(18,33,19,34)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5557);