Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C2×C4 |
| direct product | C103×#8252 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C103 | C22⋊C4 |
| commutator | C2×C4 | C2×C42×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| socle | C22×C103 | C4×C22⋊C4 |
| prime radical | C23×C103 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C103 ◁ C2×C4×C8×C103 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2,5,6)(3,4,7,8)(9,10,13,18,14,19,22,17)(11,12,16,23,21,24,20,15)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,4,6,5,7,8,2)(9,11,13,16,14,21,22,20)(10,15,18,12,19,23,17,24);(9,12,22,15,14,24,13,23)(10,16,17,11,19,20,18,21)7828 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 1240);