Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24×C4 |
| direct product | C79×#25578 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C4×D4×C79 ◁ G | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 107 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,20)(15,22,24,18)(16,21,26,19)(23,28,25,27)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(6,9)(10,11)(13,15)(14,18)(16,23)(17,24)(19,28)(20,22)(21,27)(25,26);(1,2,3,4)(15,23)(16,26)(18,27)(19,21)(22,28)(24,25);(5,7,8,12)(6,10,9,11)(13,16,17,26)(14,19,20,21)(15,23,24,25)(18,28,22,27)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 18566);