Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#13103 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1898 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| prime radical | C79×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×2−1+4×C79 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,10,7,11,24,3)(4,13,16,26,17,28,23,8)(5,18,21,19,22,9,12,14)(15,31,32,27,20,30,29,25)(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);(1,3,12,9)(2,5,19,24)(4,13,29,27)(6,11,22,14)(7,10,21,18)(8,20,30,16)(15,31,23,26)(17,28,32,25);(1,4,7,17)(2,8,11,26)(3,13,10,28)(5,20,22,15)(6,23,24,16)(9,27,18,25)(12,29,21,32)(14,31,19,30)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6091);