Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#17840 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | C22×D4 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C42 | C24×C103 |
| socle | C22×C103 | C22×C4⋊C4 |
| prime radical | C2×#41×C103 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C4×#41×C103 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 135 points:
(2,6)(4,7)(8,13)(9,14)(10,16)(12,15)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23)(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);(1,2,7,14,5,9,4,6)(3,10,15,13,11,8,12,16)(17,19,21,29,22,27,20,28)(18,23,30,32,25,31,24,26);(1,3)(2,8)(4,12)(5,11)(6,13)(7,15)(9,10)(14,16)(19,27)(23,31)(26,32)(28,29);(3,11)(8,10)(12,15)(13,16)5974 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 10828);