Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #459 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1802 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,5,12,7,13,6,10)(3,14,16,8,18,25,17,11)(4,19,20,24,22,32,23,29)(9,15,27,30,28,31,26,21)(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);(1,3,5,16,7,18,6,17)(2,8,10,14,13,11,12,25)(4,15,20,30,22,31,23,21)(9,24,26,19,28,29,27,32);(1,4)(2,9)(3,15)(5,20)(6,23)(7,22)(8,24)(10,26)(11,29)(12,27)(13,28)(14,19)(16,30)(17,21)(18,31)(25,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5810);