Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12981 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #120 |
| commutator | C23 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1869 |
| prime radical | #525×C79 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ G | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,4,9,5,10,16,24,6,8,17,21,18,22,29,31)(3,11,13,23,14,25,27,32,12,7,15,19,26,20,28,30)(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);(1,2,4,9,5,10,16,24,6,8,17,21,18,22,29,31)(3,7,15,23,14,20,28,32,12,11,13,19,26,25,27,30);(1,3)(2,7)(4,15)(5,14)(6,12)(8,11)(9,23)(10,20)(13,17)(16,28)(18,26)(19,21)(22,25)(24,32)(27,29)(30,31)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5969);