Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12963 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| socle | C2×C79 | #1867 |
| prime radical | #525×C79 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C2×C4×C79 ◁ C2×#41×C79 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,11,7,12,18,28)(3,13,17,23,14,30,32,22)(4,10,5,9,19,26,21,27)(8,20,24,31,25,16,29,15)(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);(1,3,6,17,7,14,18,32)(2,8,11,24,12,25,28,29)(4,15,5,20,19,31,21,16)(9,23,26,30,27,22,10,13);(2,9)(3,14)(8,22)(10,28)(11,26)(12,27)(13,24)(15,31)(16,20)(17,32)(23,25)(29,30)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5951);