Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C79×#2245 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C79 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(3,4)(5,7)(6,8)(9,10)(11,12,18,24)(13,20,28,39)(14,19,30,37)(15,22,32,34)(16,21,25,40)(17,33,31,23)(26,38,41,36)(27,42,29,35);(3,5,6,9)(4,7,8,10)(11,13,25,41,17,27,15,14,18,28,16,26,31,29,32,30)(12,19,34,42,23,36,21,20,24,37,22,35,33,38,40,39);(4,8)(7,10)(11,14,15,29,31,41,25,28,18,30,32,27,17,26,16,13)(12,20,21,38,33,42,34,37,24,39,40,36,23,35,22,19)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26688);