Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#10793 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C2×C103 | C22⋊#95 |
| prime radical | C22×#41×C103 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×M16×C103 ◁ G | |
1164 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)(3,8,14,23)(4,16)(5,12)(6,11)(7,10)(9,18)(13,27,29,31)(15,25,19,22)(17,26)(20,24)(21,30,32,28)(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);(1,3,6,15,7,14,5,19)(2,8,12,22,10,23,11,25)(4,13,18,30,17,29,20,28)(9,21,16,31,24,32,26,27);(1,4)(2,9)(3,13)(5,20)(6,18)(7,17)(8,21)(10,24)(11,26)(12,16)(14,29)(15,30)(19,28)(22,31)(23,32)(25,27)5974 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 3781);