Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#19215 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C103 |
| prime radical | C23×C103 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×C42×C103 ◁ G | |
582 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,22,25)(19,26,24,21)(20,27,28,32)(23,29,31,30)(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);(2,6)(7,15)(8,13)(9,16)(17,19,21,18,22,24,26,25)(20,27,29,23,28,32,30,31);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20,22,28)(18,23,25,31)(19,27,24,32)(21,29,26,30);(20,28)(23,31)(27,32)(29,30)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 12203);