Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#13234 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1964 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| prime radical | C79×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×#131×C79 ◁ G | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,10,7,11,15,3)(4,13,17,8,18,24,28,25)(5,19,21,20,22,9,12,14)(16,29,30,23,31,27,26,32)(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)(2,5,20,15)(4,13,26,23)(6,11,22,14)(7,10,21,19)(8,16,29,28)(17,25,31,27)(18,24,30,32);(1,4,15,28,7,18,6,17)(2,8,10,24,11,25,3,13)(5,16,21,30,22,31,12,26)(9,23,20,29,19,32,14,27)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6222);