Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#11368 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #124 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #400 |
| prime radical | C24×C103 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2)(3,4)(5,6,11,14,10,15,9,24)(7,16,18,12,19,28,25,27)(8,13,21,31,22,32,23,30)(17,29,20,26,35,33,34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(2,4)(5,7,11,18,10,19,9,25)(6,12,14,28,15,27,24,16)(8,20,23,17,22,34,21,35)(13,29,30,36,32,33,31,26);(2,4)(5,8,11,23,10,22,9,21)(6,13,14,30,15,32,24,31)(7,17,18,34,19,35,25,20)(12,26,28,29,27,36,16,33)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 4356);