Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×#7456 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #106 |
| commutator | C8 | C2×C16×C79 |
| Frattini, cosocle | C4×C16 | C22×C79 |
| socle | C2×C79 | #1367 |
| prime radical | C22×C79 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C2×C16×C79 ◁ C4×C16×C79 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,5,11,7,13,17,28,6,14,23,31,24,30,32,25,8,15,3,16,4,20,19,9,22,12,18,26,21,10,29,27)(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);(1,3,4,17,6,18,21,32,8,5,7,19,22,23,24,29)(2,9,10,16,12,25,20,31,15,28,30,11,14,27,13,26)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 444);