Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12563 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #106 |
| commutator | C8 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×#106 |
| 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:
(5,6)(7,18,21,30)(8,13)(9,26)(10,27)(11,17)(12,20,29,22)(14,23)(15,24)(16,25)(19,32,35,33)(28,34,31,36)(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)(5,7,10,22,8,19,24,36,11,21,9,20,25,35,23,34)(6,12,15,30,16,31,27,32,17,29,14,18,13,28,26,33);(5,8,11,25)(6,13,17,16)(7,19,21,35)(9,23,10,24)(12,28,29,31)(14,27,15,26)(18,32,30,33)(20,34,22,36)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5551);