Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33689 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | #575×C103 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×C4×C103 ◁ C4×M16×C103 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,6,10)(3,11,13,29)(4,16,17,32)(5,21,19,9)(7,22,24,14)(8,26,27,20)(12,28,15,31)(18,23,30,25)(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);(4,17)(8,27)(11,29)(14,22)(16,32)(18,30)(20,26)(28,31);(1,3)(2,7)(4,18)(5,12)(6,13)(8,28)(9,23)(10,24)(11,16)(14,20)(15,19)(17,30)(21,25)(22,26)(27,31)(29,32);(3,12,13,15)(4,17)(7,23,24,25)(8,27)(11,28,29,31)(14,18,22,30)(16,32)(20,26)3502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26677);