Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#29270 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| socle | C2×C103 | #2580 |
| prime radical | C2×#135×C103 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,12)(4,16)(5,19)(6,11)(7,14)(8,15)(9,17)(10,20)(13,26)(18,30)(21,28)(22,29)(23,32)(24,31)(25,27)(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)(2,7,11,22)(4,13)(5,14)(6,15)(8,10,12,19)(9,21,24,26)(16,25,30,23)(17,27)(18,28)(20,29)(31,32);(1,3)(2,8,11,12)(4,13)(5,14)(6,15)(7,19,22,10)(9,23,24,25)(16,21,30,26)(17,27)(18,28)(20,29)(31,32);(1,4,5,17,6,18,20,31)(2,9,10,16,11,24,19,30)(3,13,14,27,15,28,29,32)(7,21,12,25,22,26,8,23)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 22258);