Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33692 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×C4×C103 ◁ #405×C103 ◁ G | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(3,11)(4,15)(5,19)(7,20)(9,24)(10,23)(12,17)(13,28)(14,29)(16,30)(18,31)(21,32)(22,27)(25,26)(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);(1,2,6,8)(3,7,13,22)(4,16,18,9)(5,10,19,23)(11,27,28,20)(12,26,14,21)(15,30,31,24)(17,32,29,25);(1,3)(2,7)(4,17)(5,12)(6,13)(8,22)(9,25)(10,26)(11,15)(14,19)(16,32)(18,29)(20,24)(21,23)(27,30)(28,31);(2,8)(3,12,13,14)(4,18)(7,21,22,26)(10,23)(11,17,28,29)(15,31)(20,32,27,25)3502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26680);