Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C25 |
| direct product | C2×C103×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #106 |
| commutator | C8 | C23×C4×C103 |
| Frattini, cosocle | C2×C8 | C24×C103 |
| socle | C22×C103 | C2×#106 |
| prime radical | C24×C103 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ C23×C8×C103 ◁ G | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 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)(5,14,17,28)(6,11)(7,20)(8,23)(9,13)(10,16,25,30)(12,21)(15,27,18,32)(19,26)(22,33)(24,29,34,31);(3,5,7,16,8,18,21,31,9,17,22,30,6,15,19,29)(4,10,12,14,11,24,20,27,13,25,26,28,23,34,33,32);(3,6,9,8)(4,11,13,23)(5,15,17,18)(7,19,22,21)(10,24,25,34)(12,20,26,33)(14,27,28,32)(16,29,30,31)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26391);