Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#33370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | C22×#99 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C2×Q8 | C24×C103 |
| prime radical | C103×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×D4×C103 ◁ C2×2−1+4×C103 ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,5,10,6,11,18,25)(3,8,14,23,12,21,27,31)(4,9,16,24,15,7,17,20)(13,22,28,32,26,19,29,30)(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);(2,7)(4,15)(8,19)(9,11)(10,24)(13,26)(16,17)(20,25)(21,22)(23,32)(28,29)(30,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,23)(11,21)(15,26)(16,28)(17,29)(18,27)(20,30)(24,32)(25,31);(3,12)(8,21)(13,26)(14,27)(19,22)(23,31)(28,29)(30,32)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26358);