Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#17841 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | C22×D4 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C42 | C24×C103 |
| socle | C22×C103 | C22×C4⋊C4 |
| prime radical | C2×#41×C103 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C4×#41×C103 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,10)(4,8)(5,15)(6,9)(7,13)(11,12)(14,16)(17,18,21,25,22,26,20,24)(19,27,30,31,28,32,29,23)(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);(1,3,4,12,6,14,5,13)(2,7,15,16,9,11,8,10)(17,19,21,30,22,28,20,29)(18,23,24,32,26,31,25,27);(1,2)(3,11)(4,15)(5,8)(6,9)(7,14)(10,12)(13,16)(19,28)(23,31)(27,32)(29,30);(1,4,6,5)(2,8,9,15)(3,12,14,13)(7,10,11,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,32,31,27)5974 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 10829);