Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C103 · Schur multiplier (H2) = C2×C4 |
| direct product | C103×#7490 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C42×C103 |
| Frattini, cosocle | #532 | C22×C103 |
| socle | C2×C103 | #1390 |
| prime radical | #532×C103 | C22 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C2×C4×C103 ◁ C22×C4×C103 ◁ #532×C103 ◁ G | |
274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 2 generators on 135 points:
(1,2,4,9)(3,13,15,7)(5,10,20,27)(6,12,22,29)(8,23,11,21)(14,30,17,28)(16,31,19,24)(18,26,32,25)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,3,5,16,6,18,23,17)(2,7,10,24,12,25,11,30)(4,19,20,32,22,14,21,15)(8,26,9,28,27,13,29,31)3502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 478);