Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#11485 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #99 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C2×M16 | C23×C79 |
| socle | C2×C79 | #1459 |
| prime radical | #532×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C2×C8×C79 ◁ C2×#131×C79 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,11,5,10,19,23,7,12,22,31,20,28,21,30)(3,13,4,9,16,8,17,24,15,25,18,27,14,26,32,29)(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)(2,8)(4,6)(5,16)(7,15)(9,23)(10,25)(11,29)(12,26)(13,28)(14,20)(17,19)(18,22)(21,32)(24,31)(27,30);(1,4,5,17,7,18,20,32)(2,9,10,24,12,27,28,29)(3,6,16,19,15,22,14,21)(8,23,25,31,26,30,13,11)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 4473);