Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#33347 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | C2×#99 |
| commutator | C23 | C23×C4×C103 |
| Frattini, cosocle | C2×D4 | C24×C103 |
| socle | C2×C103 | C22×#99 |
| prime radical | C103×#3449 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ C22×#41×C103 ◁ G | |
1236 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)(3,11)(4,8)(5,17,18,9)(6,10)(7,14,19,26)(12,23)(13,24)(15,28,29,21)(16,22)(20,27,30,32)(25,31)(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)(2,7)(4,13)(5,14)(6,12)(8,20)(9,11)(10,19)(15,27)(16,25)(17,23)(18,26)(21,24)(22,30)(28,31)(29,32);(3,12)(7,19)(11,23)(13,25)(14,26)(20,30)(24,31)(27,32);(1,4,6,16)(2,8,10,22)(3,13,12,25)(5,15,18,29)(7,20,19,30)(9,21,17,28)(11,24,23,31)(14,27,26,32)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26335);