Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C23×C4 |
| direct product | C103×#8452 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C103 | #120 |
| commutator | C42 | C22×C4×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| prime radical | C2×#555×C103 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C2×C42×C103 ◁ G | |
868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 3 generators on 123 points:
(1,2)(3,4)(5,6,10,11,8,7,9,12)(14,15)(16,17)(19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,4)(9,10)(11,12)(13,14,18,20)(15,16,19,17);(5,7,8,6)(9,11,10,12)(13,15)(14,16)(17,20)(18,19)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 1440);