Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#12646 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #102 |
| commutator | C8 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | C4×#102 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,10,7,13,23,12)(3,14,17,31,18,25,32,27)(4,9,19,11,20,30,5,21)(8,24,26,22,29,15,28,16)(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);(1,3,4,15,6,17,19,16,7,18,20,24,23,32,5,22)(2,8,11,27,12,28,9,25,13,29,21,31,10,26,30,14);(1,4,6,19,7,20,23,5)(2,9,10,11,13,30,12,21)(3,15,17,16,18,24,32,22)(8,25,26,27,29,14,28,31)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5634);