Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C25 |
| direct product | C2×C103×#1534 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C103 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C42 | C24×C103 |
| prime radical | C24×C103 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C22×C42×C103 ◁ G | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(3,4,6,10,7,9,15,17)(5,11,12,18,13,8,14,16)(19,20)(21,27,29,34)(22,26)(23,25)(24,28,32,30)(31,33);(3,5,7,13)(4,8,9,11)(6,14,15,12)(10,18,17,16)(19,21)(20,24)(22,30)(23,29)(25,32)(26,34)(27,33)(28,31);(20,25)(24,32)(26,33)(27,34)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 10645);