Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C89×#3156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C22×#135×C89 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×#117×C89 ◁ G | |
3150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 127 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(4,6)(7,8,12,16)(9,17,21,31)(10,14,22,28)(11,24,18,15)(13,25,27,20)(19,30,33,38)(23,35,32,29)(26,36,37,34);(7,9,18,25,12,21,11,20)(8,13,24,31,16,27,15,17)(10,19,32,36,22,33,23,34)(14,26,35,38,28,37,29,30);(3,4,5,6)(7,10)(8,14)(9,19)(11,23)(12,22)(13,26)(15,29)(16,28)(17,30)(18,32)(20,34)(21,33)(24,35)(25,36)(27,37)(31,38);(10,22)(14,28)(19,33)(23,32)(26,37)(29,35)(30,38)(34,36)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53630);