Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C103 · Schur multiplier (H2) = C23×C4 |
| direct product | C103×#7872 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C103 | D4 |
| commutator | C4 | C2×C4×C8×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| socle | C23×C103 | C2×C4⋊C4 |
| prime radical | C24×C103 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C103 ◁ C22×C42×C103 ◁ G | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 3 generators on 131 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,28)(26,27)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(2,4)(13,15,17,25,18,26,16,24)(14,19,20,27,22,28,21,23);(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,27,28,23)11536 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 860);