Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C89 · Schur multiplier (H2) = C22 |
| direct product | C89×#11643 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C89 |
| Frattini, cosocle | C4×C8 | C23×C89 |
| socle | C22×C89 | C2×C22⋊C8 |
| prime radical | C23×C89 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×C4×C8×C89 ◁ G | |
288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,21,27,22,28,36,45,23,25,37,41,20,26,34,43)(19,29,31,46,30,39,47,44,32,42,35,48,33,24,38,40)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,39)(26,42)(27,44)(28,29)(31,37)(34,47)(36,38)(40,41)(43,46)(45,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,25)(24,39)(26,28)(27,41)(29,42)(40,44)(43,45)(46,48)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4631);