Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C103×#2190 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×C4×C103 ◁ C4×M16×C103 ◁ G | |
526 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)(3,8)(4,14)(5,18)(6,22)(7,11)(9,17)(10,19)(12,29)(13,28)(15,27)(16,31)(20,23)(21,26)(24,30)(25,32)(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);(1,3,7,6)(2,8,11,22)(4,12,16,15)(5,19,21,13)(9,23,24,25)(10,26,28,18)(14,27,31,29)(17,32,30,20);(1,4)(2,9)(3,12)(5,20)(6,15)(7,16)(8,23)(10,27)(11,24)(13,30)(14,18)(17,19)(21,32)(22,25)(26,31)(28,29);(4,15,16,12)(5,21)(9,23,24,25)(10,28)(13,19)(14,27,31,29)(17,20,30,32)(18,26)3502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26678);