Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#12713 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1781 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| prime radical | C79×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×H(5,𝔽2)×C79 ◁ G | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,12)(3,14,17,23)(4,19,20,9)(5,11,21,10)(7,13,22,27)(8,16,25,31)(15,26,30,28)(18,29,32,24)(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);(1,3,7,18)(2,8,13,26)(4,15,21,31)(5,16,20,30)(6,17,22,32)(9,23,11,24)(10,14,19,29)(12,28,27,25);(2,9)(5,20)(6,22)(8,23)(10,12)(11,13)(14,28)(16,30)(17,32)(19,27)(24,26)(25,29)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5701);