Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C22×C4 |
| direct product | C103×#13232 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #1964 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C2×#41 | C23×C103 |
| prime radical | #535×C103 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×C4×C103 ◁ C2×#131×C103 ◁ G | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2,7,12,8,13,6,25)(3,14,18,21,19,29,17,32)(4,20,9,27,23,16,22,10)(5,11,15,28,24,31,26,30)(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);(1,3,15,10)(2,9,28,21)(4,11,29,12)(5,16,7,18)(6,17,24,20)(8,19,26,27)(13,22,30,32)(14,25,23,31);(1,4,6,22,8,23,7,9)(2,10,12,20,13,27,25,16)(3,11,17,30,19,31,18,28)(5,21,15,29,24,32,26,14)3502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 6220);