Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#11369 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #127 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #399 |
| prime radical | C24×C103 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ G | |
602 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)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,24,21)(16,31,33,30)(19,26)(20,27)(22,29)(23,34,25,32)(35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(2,4)(5,7,9,21,10,18,22,17)(6,11,13,28,14,24,29,15)(8,16,19,34,20,33,35,32)(12,25,26,30,27,23,36,31);(2,4)(5,8,9,19,10,20,22,35)(6,12,13,26,14,27,29,36)(7,16,21,34,18,33,17,32)(11,23,28,31,24,25,15,30)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 4357);