Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C2×C103×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C2×#99 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C2×Q8 | C24×C103 |
| prime radical | C22×#41×C103 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C24×C103 ◁ C23×Q8×C103 ◁ G | |
1198 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);(3,4,7,11,8,12,23,30)(5,13,16,31,14,27,21,33)(6,17,19,29,20,10,22,28)(9,24,26,18,25,32,34,15);(3,5)(4,9)(6,18)(7,21)(8,14)(10,27)(11,26)(12,25)(13,17)(15,20)(16,23)(19,24)(22,32)(28,33)(29,31)(30,34);(5,14)(9,25)(13,27)(15,18)(16,21)(24,32)(26,34)(31,33)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26354);