Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#12940 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | #447 |
| commutator | C2×C8 | C22×C4×C103 |
| Frattini, cosocle | C22×C8 | C23×C103 |
| prime radical | C22×D8×C103 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ #405×C103 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2,4,3)(5,6)(7,18)(8,23)(9,13)(10,28)(11,17)(12,19)(14,24)(15,26)(16,25)(20,35)(21,30)(22,34)(27,33)(29,31)(32,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,3,4,2)(5,7)(6,12)(8,20)(9,27)(10,29)(11,22)(13,30)(14,31)(15,34)(16,35)(17,33)(18,23)(19,24)(21,25)(26,36)(28,32);(6,13)(8,24)(9,26)(10,25)(12,30)(14,17)(15,28)(16,23)(18,35)(19,20)(21,29)(27,36)(31,33)(32,34)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 5928);