Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#33351 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | C22×#99 |
| commutator | C23 | C23×C4×C103 |
| Frattini, cosocle | C2×D4 | C24×C103 |
| prime radical | C103×#3487 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×Q8×C103 ◁ C2×2−1+4×C103 ◁ G | |
1212 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)(3,13)(4,9)(5,10)(6,20,21,11)(7,12)(8,15,23,19)(14,24)(16,29,30,25)(17,26)(18,27)(22,28,32,31)(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);(1,3,7,5)(2,8,12,23)(4,14,17,18)(6,15,21,19)(9,22,26,32)(10,11,13,20)(16,28,30,31)(24,29,27,25);(1,4,7,17)(2,9,12,26)(3,14,5,18)(6,16,21,30)(8,22,23,32)(10,27,13,24)(11,25,20,29)(15,28,19,31);(1,5,7,3)(2,10,12,13)(4,14,17,18)(6,19,21,15)(8,11,23,20)(9,24,26,27)(16,28,30,31)(22,29,32,25)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26339);