Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C89 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C89×#1757 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C89 | C22⋊C4 |
| commutator | C22 | C22×C42×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C22×C89 | C4×C22⋊C4 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C89 ◁ C23×C8×C89 ◁ G | |
1474 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);(3,4,7,11,8,12,19,25)(5,13,16,27,15,9,20,22)(6,10,17,23,18,24,30,33)(14,26,29,34,28,21,31,32);(3,5)(4,9,12,13)(6,14)(7,16)(8,15)(10,21,24,26)(11,22,25,27)(17,29)(18,28)(19,20)(23,32,33,34)(30,31);(3,6,7,17,8,18,19,30)(4,10,11,23,12,24,25,33)(5,14,16,29,15,28,20,31)(9,21,22,32,13,26,27,34)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 13388);