Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#17978 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C22×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×C4×D4 |
| prime radical | #466×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,8,6,9,14,3)(4,11,13,10,12,7,15,16)(17,18)(19,26)(20,23)(21,25)(22,28)(24,30)(27,31)(29,32)(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);(4,12)(7,11)(10,16)(13,15)(17,19)(18,22)(20,27)(21,29)(23,26)(24,32)(25,31)(28,30);(1,3,5,2,6,8,14,9)(4,10,13,7,12,16,15,11);(1,4)(2,7)(3,10)(5,13)(6,12)(8,16)(9,11)(14,15)(17,20,21,30)(18,23,25,24)(19,27,29,28)(22,26,31,32)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10966);