Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C24×C4 |
| direct product | C103×#10956 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #120 |
| commutator | C2×C4 | C2×C42×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #386 |
| prime radical | C22×#41×C103 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ G | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 139 points:
(5,6)(7,16)(8,20,22,29)(9,23)(10,14)(11,15)(12,18)(13,21,30,25)(17,32,35,26)(19,27)(24,28)(31,34,36,33)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,2,3,4)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,26)(14,27)(15,28)(16,23)(20,31)(21,34)(22,35)(25,33)(29,36)(30,32);(5,8,10,25,11,22,9,21)(6,13,14,29,15,30,23,20)(7,17,18,33,19,35,24,34)(12,26,27,36,28,32,16,31)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 3944);