Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C23×C8 |
| direct product | C103×#19656 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C103 |
| prime radical | C103×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×C42×C103 ◁ G | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 135 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(18,22)(23,30)(24,29)(25,31);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(3,9)(7,14)(10,15)(12,16)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 12644);