Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#33359 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | C22×#99 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C2×D4 | C24×C103 |
| prime radical | C103×#3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C2×D4×C103 ◁ C2×H(5,𝔽2)×C103 ◁ G | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 1648 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,7)(4,8,12,6)(5,9)(10,14)(11,15,16,13)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16);(3,10)(7,14)(11,16)(13,15)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26347);