Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12988 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #386 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1869 |
| prime radical | #532×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C8×C79 ◁ G | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,7,11,5,12,15,30,6,13,24,31,23,10,29,26)(3,14,17,20,16,8,25,19,18,27,32,4,9,21,28,22)(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);(1,3,15,25,23,9,7,17,6,18,29,28,5,16,24,32)(2,8,26,20,10,14,31,22,13,21,30,4,12,27,11,19);(1,4)(2,9)(3,12)(5,22)(6,20)(7,8)(10,18)(11,25)(13,16)(14,29)(15,27)(17,26)(19,23)(21,24)(28,31)(30,32)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5976);