Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12815 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×#102 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 115 points:
(2,4)(5,6,9,14,11,15,10,25)(7,16,18,32,19,33,26,30)(8,20,22,31,24,35,23,13)(12,21,27,34,29,36,28,17)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,3,4);(5,7)(6,12)(8,21)(9,18)(10,26)(11,19)(13,30)(14,27)(15,29)(16,20)(17,23)(22,34)(24,36)(25,28)(31,32)(33,35)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5803);