Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C103 · Schur multiplier (H2) = C22 |
| direct product | Q8×C32×C103 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C103 | C22 |
| commutator | C2 | C22×C32×C103 |
| Frattini, cosocle | C2×C16 | C23×C103 |
| socle, prime radical | C22×C103 | C22×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C103 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 26368 |
A minimal faithful permutation representation — 3 generators on 143 points:
(9,10,11,15,12,16,19,25,13,17,20,26,22,28,31,35,14,18,21,27,23,29,32,36,24,30,33,37,34,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(1,2,4,6)(3,7,8,5);(1,3,4,8)(2,5,6,7)16480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 6647);