Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C79×#1572 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C2×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C23×C79 | C2×C4⋊C4 |
| prime radical | C24×C79 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C22×C42×C79 ◁ G | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(19,20)(21,28)(22,26)(23,32)(24,27)(25,30)(29,33)(31,34);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(19,21,23,30,24,31,22,29)(20,25,26,28,27,33,32,34);(5,11)(8,14)(12,17)(15,18)(19,22,24,23)(20,26,27,32)(21,29,31,30)(25,28,33,34)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10790);