Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16×C79 · Schur multiplier (H2) = C23 |
| direct product | C2×C79×#1397 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16×C79 | C22 |
| commutator | C2 | C2×C4×C16×C79 |
| Frattini, cosocle | C2×C16 | C23×C79 |
| socle, prime radical | C23×C79 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16×C79 ◁ G | |
254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(7,8,9,14,10,15,17,26,11,16,18,27,20,28,29,36,12,13,19,23,21,24,30,33,22,25,31,34,32,35,37,38);(3,4,5,6)(8,13)(14,23)(15,24)(16,25)(26,33)(27,34)(28,35)(36,38)12640 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6535);