Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#13336 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C79 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ #405×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:
(5,6)(7,18)(8,12)(9,25)(10,28)(11,17)(13,20)(14,22)(15,26)(16,24)(19,31)(21,30)(23,32)(27,35)(29,34)(33,36)(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)(6,12)(7,19)(8,22)(9,26)(10,24)(13,31)(14,16)(15,28)(17,25)(18,35)(20,36)(21,29)(30,32)(33,34);(2,4)(5,7)(6,13)(8,23)(9,27)(10,29)(11,21)(12,30)(14,31)(15,34)(16,35)(17,33)(18,25)(19,24)(20,26)(22,36)(28,32)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6324);