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×#2135 |
| 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 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ C22×#41×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);(3,4,8,13,9,14,6,11)(5,15,18,33,19,30,16,32)(7,21,20,31,23,12,24,29)(10,25,26,22,27,34,28,17);(3,5,9,19)(4,10,14,27)(6,18,8,16)(7,22,23,17)(11,28,13,26)(12,30,21,15)(20,25,24,34)(29,32,31,33);(3,6,9,8)(4,11,14,13)(5,16,19,18)(7,20,23,24)(10,26,27,28)(12,29,21,31)(15,32,30,33)(17,25,22,34)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26356);