Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | #459 |
| commutator | C2×C8 | C22×C4×C103 |
| Frattini, cosocle | C22×C8 | C23×C103 |
| socle | C2×C103 | #1802 |
| prime radical | C2×#135×C103 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ C2×C4×C8×C103 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2,7,11,8,13,5,21)(3,12,6,22,16,29,15,9)(4,17,19,32,20,24,18,28)(10,23,25,31,27,30,26,14)(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);(1,3,5,15,8,16,7,6)(2,9,11,12,13,22,21,29)(4,14,18,30,20,31,19,23)(10,24,25,28,27,17,26,32);(1,4)(2,10)(3,14)(5,18)(6,23)(7,19)(8,20)(9,24)(11,25)(12,28)(13,27)(15,30)(16,31)(17,22)(21,26)(29,32)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 5810);