Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C103 · Schur multiplier (H2) = C2 |
| direct product | C103×#7521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #1402 |
| commutator | C4⋊Q8 | C2×C4×C103 |
| Frattini, cosocle | #494 | C22×C103 |
| prime radical | #555×C103 | C4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C22×C103 ◁ C2×C4×C103 ◁ C42×C103 ◁ #494×C103 ◁ G | |
148 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 30 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:
(1,2,4,9,5,11,17,29)(3,12,14,26,13,23,32,27)(6,10,21,30,19,31,18,28)(7,16,22,20,8,24,25,15)(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);(2,7)(3,13)(8,11)(9,26)(10,12)(14,32)(15,20)(16,24)(22,28)(23,31)(25,30)(27,29)2266 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 509);