Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33594 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×#459 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
696 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)(3,14)(4,19)(5,10)(6,12)(7,13)(8,17)(9,22)(11,16)(15,29)(18,25)(20,28)(21,24)(23,32)(26,31)(27,30)(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);(1,3,6,17,7,18,5,16)(2,8,10,14,13,11,12,25)(4,15,21,30,23,31,20,22)(9,24,26,19,27,28,29,32);(1,4)(2,9)(3,15)(5,20)(6,21)(7,23)(8,24)(10,26)(11,28)(12,29)(13,27)(14,19)(16,22)(17,30)(18,31)(25,32);(1,5,7,6)(2,10,13,12)(3,16,18,17)(4,20,23,21)(8,14,11,25)(9,26,27,29)(15,22,31,30)(19,28,32,24)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26582);