Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C24×C4 |
| direct product | C103×#22284 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #2328 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C103×#3154 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C22×C4×C103 ◁ G | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,5)(4,17,21,27)(6,12)(7,13)(8,15)(9,20,28,31)(10,22)(11,24)(14,29,30,25)(16,23)(18,19,26,32)(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);(1,3,6,15,7,16,24,10)(2,8,11,5,13,22,12,23)(4,18,31,14,21,26,20,30)(9,19,17,25,28,32,27,29);(1,4,7,21)(2,9)(3,14)(5,19,23,32)(6,20,24,31)(8,25,22,29)(10,26)(11,27)(12,17)(13,28)(15,18)(16,30);(1,5,6,22,7,23,24,8)(2,10,11,3,13,15,12,16)(4,19,20,29,21,32,31,25)(9,26,27,14,28,18,17,30)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 15272);