Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33656 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| socle | C2×C103 | C2×#481 |
| prime radical | C103×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ C22×M16×C103 ◁ G | |
556 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)(3,12)(4,9)(5,17)(6,21)(7,11)(8,15)(10,20)(13,27)(14,28)(16,26)(18,32)(19,25)(22,29)(23,30)(24,31)(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);(5,18)(10,26)(12,28)(15,31)(16,20)(17,32)(19,30)(27,29);(1,3,6,22,7,14,4,13)(2,8,9,23,11,24,21,25)(5,19,20,31,18,30,16,15)(10,27,17,28,26,29,32,12);(1,4,7,6)(2,9,11,21)(3,13,14,22)(5,16,18,20)(8,23,24,25)(10,17,26,32)(12,27,28,29)(15,30,31,19)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26644);