Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C2×C89×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C2×#135×C89 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C22×C4×C89 ◁ C2×C4×C8×C89 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(3,4,9,14)(5,15,18,13)(6,19)(7,23,16,12)(8,10,25,27)(11,28)(17,31)(20,33)(21,32)(22,26)(24,29)(30,34);(3,5,16,25,9,18,7,8)(4,10,23,13,14,27,12,15)(6,17,21,34,22,28,20,24)(11,26,30,32,31,19,29,33);(3,6)(4,11)(5,17)(7,20)(8,24)(9,22)(10,26)(12,29)(13,32)(14,31)(15,33)(16,21)(18,28)(19,27)(23,30)(25,34)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26624);