Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C103 · Schur multiplier (H2) = C22 |
| direct product | C2×C103×#1398 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C103 | C42 |
| commutator | C4 | C2×C4×C8×C103 |
| Frattini, cosocle | C2×C16 | C23×C103 |
| socle | C22×C103 | #346 |
| prime radical | C23×C103 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C103 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(3,4,5,10,6,12,15,25,7,13,16,23,17,27,29,33,8,11,14,24,18,26,28,34,19,9,20,21,30,22,31,32);(4,9,11,13)(5,14)(10,23,24,21)(12,22,26,27)(15,28)(16,20)(25,33,34,32)(29,31)9064 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 6538);