Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#9056 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #120 |
| commutator | C42 | C22×C4×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| socle | C22×C103 | #386 |
| prime radical | C23×C103 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C2×C42×C103 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 135 points:
(2,6,8,11)(3,9,10,12)(17,18)(19,27,30,31)(20,24,23,29)(21,25)(22,26)(28,32)(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);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,28,29,22,30,21,24)(18,23,32,27,26,20,25,31);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20,22,23)(18,24,26,29)(19,25,30,32)(21,27,28,31)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 2044);