Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#33350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | C22×#99 |
| commutator | C23 | C23×C4×C103 |
| Frattini, cosocle | C2×D4 | C24×C103 |
| prime radical | C103×#3485 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×D4×C103 ◁ C2×H(5,𝔽2)×C103 ◁ G | |
1240 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)(3,11)(4,8)(5,17,18,9)(6,10)(7,13,20,27)(12,23)(14,24)(15,21)(16,29,28,22)(19,26,30,32)(25,31)(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);(1,3,6,14)(2,7,10,20)(4,12,15,25)(5,13,18,27)(8,19,21,30)(9,11,17,24)(16,26,28,32)(22,23,29,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,21)(11,23)(13,26)(14,25)(17,29)(18,28)(20,30)(24,31)(27,32);(4,15)(8,21)(12,25)(16,28)(19,30)(22,29)(23,31)(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, 26338);