Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#12567 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #102 |
| commutator | C8 | C2×C42×C103 |
| Frattini, cosocle | C22×C8 | C23×C103 |
| socle | C22×C103 | C2×#102 |
| prime radical | C24×C103 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ C23×C8×C103 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2,3,4)(5,6)(7,18)(8,14)(9,25)(10,15)(11,28)(12,17)(13,20)(16,27)(19,30)(21,29)(22,36)(23,32)(24,33)(26,31)(34,35)(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);(5,7,10,21,11,22,27,20,12,23,24,35,8,19,9,26)(6,13,16,30,14,29,15,32,17,31,25,36,28,34,33,18);(5,8,12,11)(6,14,17,28)(7,19,23,22)(9,24,27,10)(13,29,31,34)(15,25,33,16)(18,30,32,36)(20,21,26,35)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 5555);