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) = C28 |
| direct product | C2×C79×#3396 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | C2×#493×C79 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C4×Q8×C79 ◁ G | |
2086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 105 points:
(1,2)(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);(4,8)(9,10)(11,12,16,19)(13,17,15,23)(14,21,22,25)(18,24,26,20);(3,4,7,8)(5,9,6,10)(11,13,16,15)(12,17,19,23)(14,20,22,24)(18,25,26,21);(3,5,7,6)(4,9,8,10)(11,14,16,22)(12,18,19,26)(13,20,15,24)(17,25,23,21);(3,6,7,5)(4,9,8,10)5530 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54380);