Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C2×C103×#2214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C2×#135×C103 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ C2×C4×C8×C103 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2)(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);(3,4,10,16)(5,14,19,25)(6,20,22,33)(7,15,9,13)(8,11,18,26)(12,27,29,17)(21,34,24,30)(23,31,32,28);(3,5,9,18,10,19,7,8)(4,11,15,25,16,26,13,14)(6,17,24,32,22,27,21,23)(12,20,31,34,29,33,28,30);(3,6)(4,12)(5,17)(7,21)(8,23)(9,24)(10,22)(11,20)(13,28)(14,30)(15,31)(16,29)(18,32)(19,27)(25,34)(26,33)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26581);