Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C29 |
| direct product | C89×#62636 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | C23×D8 |
| commutator, Frattini, cosocle | C8 | C25×C89 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×#41×C89 ◁ C2×#131×C89 ◁ G | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 5 generators on 121 points:
(1,2)(3,8)(4,7)(5,18)(6,9)(10,15)(11,16)(12,22)(13,19)(14,28)(17,20)(21,29)(23,27)(24,26)(25,30)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(2,7)(4,15)(5,16)(8,19)(9,20)(10,11)(13,27)(14,26)(17,29)(18,21)(22,30)(23,24)(25,32)(28,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,23)(11,24)(15,27)(16,26)(17,25)(18,28)(20,30)(21,31)(29,32);(3,12)(8,22)(13,25)(14,26)(19,30)(23,31)(24,28)(27,32);(2,9)(7,20)(8,22)(10,21)(11,18)(19,30)(23,31)(24,28)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 55624);