Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#29233 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| socle | C2×C103 | #2572 |
| prime radical | #575×C103 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,6,11,7,12,5,10)(3,13,17,24,15,28,16,8)(4,18,20,27,21,9,22,26)(14,25,19,32,30,29,31,23)(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)(2,8,12,24)(4,19,21,31)(5,16)(6,17)(7,15)(9,25)(10,28,11,13)(14,20,30,22)(18,29)(23,27)(26,32);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,23)(10,26)(11,27)(12,18)(13,29)(15,30)(16,19)(17,31)(24,32)(25,28);(2,10,12,11)(3,15)(8,13,24,28)(9,26,18,27)(14,30)(16,17)(19,31)(23,29,32,25)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 22221);