Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#8124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #95 |
| commutator | C23 | C2×C42×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| socle | C22×C103 | C2×#95 |
| prime radical | C24×C103 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C22×C42×C103 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2,4,8,6,9,5,10)(3,11,13,15,14,7,12,16)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30)(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);(1,3,5,13,6,14,4,12)(2,7,10,16,9,11,8,15)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,11,16)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 1112);