Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#12780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1790 |
| commutator | C2×Q8 | C22×C4×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| prime radical | C79×#3551 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×#41×C79 ◁ G | |
756 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)(3,12)(4,9,18,28)(5,10)(6,11)(7,14,24,13)(8,17,26,20)(15,22)(16,27)(19,32)(21,25)(23,31,29,30)(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)(2,7)(4,13)(5,19)(6,16)(8,22)(9,27)(10,23)(11,24)(12,28)(14,18)(15,21)(17,31)(20,30)(25,29)(26,32);(1,4,6,18)(2,8,11,26)(3,13,16,14)(5,20,21,17)(7,22,24,32)(9,10,28,25)(12,29,27,23)(15,31,19,30)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5768);