Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C22×C4 |
| direct product | C103×#7878 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C103 | C2×D4 |
| commutator | C2×C4 | C2×C42×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| socle | C23×C103 | C2×C4⋊C4 |
| prime radical | C24×C103 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C103 ◁ C22×C42×C103 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(2,4)(5,7,9,17,10,18,8,16)(6,11,12,19,14,20,13,15)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(23,29)(26,32)(30,35)(33,36)7828 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 866);