Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#29427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C103 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C24×C103 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C2×#117×C103 ◁ G | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,20,29)(21,26,22,31)(25,28,32,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,24,27,29)(21,30,22,28)(25,31,32,26);(11,16)(13,15)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 22415);