Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#13397 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C103 |
| prime radical | C2×#41×C103 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C22×C4×C103 ◁ C2×#131×C103 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2,8,14)(3,9,18,34)(4,19,21,53)(5,25,29,41)(6,13,7,12)(10,35,37,22)(11,27,42,28)(15,44,45,64)(16,47,50,63)(17,43,30,33)(20,55,51,52)(23,54,24,40)(26,39,57,38)(31,59,60,46)(32,58,61,49)(36,56,62,48)(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)(2,9)(4,20)(5,16)(6,30)(7,17)(8,18)(10,36)(11,32)(12,43)(13,33)(14,34)(15,23)(19,52)(21,51)(22,56)(24,45)(25,47)(26,46)(27,58)(28,49)(29,50)(31,38)(35,48)(37,62)(39,60)(40,44)(41,63)(42,61)(53,55)(54,64)(57,59);(1,4,7,23,8,21,6,24)(2,10,13,38,14,37,12,39)(3,15,17,51,18,45,30,20)(5,26,28,35,29,57,27,22)(9,31,33,62,34,60,43,36)(11,40,25,53,42,54,41,19)(16,48,49,59,50,56,58,46)(32,55,47,64,61,52,63,44)3502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 6385);