Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#22694 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | D42 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | D4×C4⋊C4 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ G | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,7,12,8,14,6,13)(3,15,17,9,18,27,24,28)(4,19,21,10,22,29,20,30)(5,11,16,31,23,32,25,26)(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)(2,9)(4,16)(5,20)(6,24)(7,17)(8,18)(10,26)(11,29)(12,27)(13,15)(14,28)(19,32)(21,23)(22,25)(30,31);(1,4,7,21,8,22,6,20)(2,10,13,19,14,30,12,29)(3,16,17,23,18,25,24,5)(9,26,15,32,28,31,27,11);(1,5)(2,11)(3,4)(6,25)(7,16)(8,23)(9,10)(12,31)(13,26)(14,32)(15,19)(17,21)(18,22)(20,24)(27,29)(28,30)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15682);