Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#9330 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | #386 |
| commutator | C42 | C22×C4×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| prime radical | #532×C103 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×C42×C103 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2,4,6,5,3,8,7)(9,10,13,17,14,18,12,16)(11,19,20,23,22,24,21,15)(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)(2,5)(4,6)(7,8)(9,11)(10,15,18,23)(12,21)(13,20)(14,22)(16,24,17,19);(1,2)(3,5)(4,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,23,24)3811 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 2318);