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) = C2×C42 |
| direct product | C79×#12590 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8×C79 | C22 |
| commutator | C2 | C2×C4×C16×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle, prime radical | C23×C79 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8×C79 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 107 points:
(5,6)(7,10)(8,9)(11,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,2,3,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)12640 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5578);