Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C4 |
| direct product | C103×#7673 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C103 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C43×C103 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 26368 |
A minimal faithful permutation representation — 3 generators on 135 points:
(2,5)(3,7)(6,8)(10,14,15,16)(17,18,20,24,21,23,29,31)(19,25,27,32,26,22,28,30)(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);(1,2,4,6)(3,8,7,5)(9,10)(11,14)(12,16)(13,15)(17,19,21,26)(18,22,23,25)(20,28,29,27)(24,32,31,30);(9,11,13,12)(10,14,15,16)(19,26)(22,25)(27,28)(30,32)5974 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 661);