Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#33349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | C22×#99 |
| commutator | C23 | C23×C4×C103 |
| Frattini, cosocle | C2×D4 | C24×C103 |
| prime radical | C103×#3486 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×Q8×C103 ◁ C2×2−1+4×C103 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,8)(4,9)(5,17,18,7)(6,10)(11,21)(12,26,27,19)(13,22)(14,23)(15,29,30,20)(16,24)(25,32,28,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(2,7)(8,19)(9,20)(10,17)(21,31)(22,26)(23,32)(24,29);(1,3,6,13)(2,8,10,22)(4,14,16,11)(5,12,18,27)(7,19,17,26)(9,23,24,21)(15,28,30,25)(20,32,29,31);(1,4,6,16)(2,9,10,24)(3,11,13,14)(5,15,18,30)(7,20,17,29)(8,21,22,23)(12,25,27,28)(19,31,26,32)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26337);