Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×#7453 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1365 |
| commutator | C2×C16 | C2×C4×C79 |
| Frattini, cosocle | C4×C16 | C22×C79 |
| prime radical | C22×C79 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C4×C8×C79 ◁ C4×C16×C79 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,4,9,6,23,19,32,5,22,20,30,16,31,18,29,8,13,21,27,25,12,3,14,7,11,15,26,24,28,17,10)(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);(2,9,13,27)(3,15)(4,18)(5,7)(6,24)(10,12,29,23)(11,30,22,26)(14,28,32,31)(16,25)(17,21)(19,20)2923 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 441);