Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C22 |
| direct product | C103×#8877 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #120 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| socle | C22×C103 | #385 |
| prime radical | #121×C103 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C2×C42×C103 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18,21,26,23,20,22,25)(19,27,30,28,31,32,29,24)(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);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,15,11,13)(17,19,21,30,23,31,22,29)(18,24,26,27,20,28,25,32);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,15,11,13)(17,20,23,18)(19,28,31,24)(21,25,22,26)(27,30,32,29)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 1865);