Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C29 |
| direct product | C2×C79×#3283 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C4×D4×C79 ◁ G | |
2806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 101 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(3,4)(5,6)(7,8,12,15)(9,16,19,18)(10,14,11,20)(13,21,22,17);(7,9)(8,13)(10,17)(11,21)(12,19)(14,18)(15,22)(16,20);(4,6)(7,10,12,11)(8,14,15,20)(9,17,19,21)(13,18,22,16);(7,8)(9,18)(10,14)(11,20)(12,15)(13,21)(16,19)(17,22)5530 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54031);