Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#18519 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C103 |
| socle | C22×C103 | C2×#120 |
| prime radical | C2×#41×C103 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C2×C42×C103 ◁ G | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,23,28)(19,25,30,31)(20,24,29,32)(21,26,22,27)(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);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,23,30)(18,24,28,32)(20,21,29,22)(25,27,31,26);(17,20)(18,25)(19,21)(22,30)(23,29)(24,27)(26,32)(28,31);(17,21,23,22)(18,26,28,27)(19,29,30,20)(24,25,32,31)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 11507);