Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C89 · Schur multiplier (H2) = C26 |
| direct product | C89×#10787 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #386 |
| commutator | C2×C4 | C2×C42×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C2×C89 | C22⋊#95 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C23×C4×C89 ◁ G | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,5,11,7,10,6,20)(3,8,15,23,13,24,14,22)(4,16,17,25,18,26,19,9)(12,27,30,21,28,31,29,32)(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);(1,3,6,15,7,13,5,14)(2,8,11,22,10,24,20,23)(4,12,17,29,18,28,19,30)(9,21,26,31,25,32,16,27);(1,4)(2,9)(3,12)(5,19)(6,17)(7,18)(8,21)(10,25)(11,26)(13,28)(14,30)(15,29)(16,20)(22,31)(23,27)(24,32)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 3775);