Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28×C4 |
| direct product | C89×#60649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 125 points:
(2,4)(5,6)(7,15)(8,13)(9,21)(10,12)(11,17)(14,22)(16,28)(18,27)(19,25)(20,33)(23,31)(24,30)(26,32)(29,35)(34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(5,7,9,17,10,18,22,31)(6,11,14,15,12,23,21,27)(8,16,20,30,19,29,32,36)(13,24,26,28,25,34,33,35);(1,2,3,4)(6,12)(11,23)(13,25)(14,21)(15,27)(24,34)(26,33)(28,35);(5,8)(6,13)(7,16)(9,20)(10,19)(11,24)(12,25)(14,26)(15,28)(17,30)(18,29)(21,33)(22,32)(23,34)(27,35)(31,36);(8,19)(13,25)(16,29)(20,32)(24,34)(26,33)(28,35)(30,36)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53637);