Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#11363 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #397 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C2×C89 | #1445 |
| prime radical | C24×C89 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C23×C4×C89 ◁ G | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,6,12,7,10,5,11)(3,8,16,24,14,25,15,23)(4,17,19,27,21,28,20,9)(13,26,30,31,29,32,18,22)(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);(1,3,5,16,7,14,6,15)(2,8,12,23,10,25,11,24)(4,18,19,29,21,30,20,13)(9,26,28,31,27,32,17,22);(1,4,5,19,7,21,6,20)(2,9,12,28,10,27,11,17)(3,13,16,18,14,29,15,30)(8,22,23,26,25,31,24,32)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4351);