Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C103 · Schur multiplier (H2) = C2×C4 |
| direct product | C103×#7475 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | #367 |
| commutator | C16 | C42×C103 |
| Frattini, cosocle | C22×C16 | C22×C103 |
| socle | C2×C103 | #1380 |
| prime radical | C23×C103 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ C22×C8×C103 ◁ C22×C16×C103 ◁ G | |
170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 2 generators on 167 points:
(1,2)(3,15,17,43)(4,11)(5,25)(6,28)(7,31)(8,14)(9,16,32,46)(10,22)(12,30)(13,29)(18,51,47,42)(19,53,49,64)(20,55,50,44)(21,38)(23,58)(24,41)(26,36)(27,37)(33,45,60,54)(34,62,61,52)(35,63,59,48)(39,57)(40,56)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,3,5,18,26,49,56,63,29,50,57,62,30,54,24,46,8,17,22,47,6,19,23,48,7,20,27,52,21,45,4,16)(2,9,11,33,38,61,37,44,31,59,58,64,28,51,10,15,14,32,41,60,12,34,39,55,13,35,40,53,36,42,25,43)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 463);