Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33572 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C79 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×#41×C79 ◁ G | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2)(3,13)(4,18)(5,9)(6,12)(7,16)(8,20)(10,21)(11,23)(14,26)(15,24)(17,29)(19,27)(22,30)(25,32)(28,31)(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)(4,14)(5,22)(6,17)(8,24)(9,28)(10,29)(11,25)(12,26)(13,18)(15,23)(16,21)(19,32)(20,31)(27,30);(1,4,6,21)(2,8,12,27)(3,14,17,16)(5,19,23,20)(7,24,26,30)(9,18,11,10)(13,25,29,28)(15,31,22,32);(1,5,6,23)(2,9,12,11)(3,15,17,22)(4,19,21,20)(7,25,26,28)(8,18,27,10)(13,24,29,30)(14,31,16,32)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26560);