Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C131 · Schur multiplier (H2) = C2×C4 |
| direct product | C131×#7482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #1390 |
| commutator | C2×Q8 | C42×C131 |
| Frattini, cosocle | #532 | C22×C131 |
| prime radical | #532×C131 | C22 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C2×C4×C131 ◁ C22×C4×C131 ◁ #532×C131 ◁ G | |
278 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 278 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) = 147 |
| transitive degree | μt(G) = 2096 |
A minimal faithful permutation representation — 2 generators on 147 points:
(1,2,4,7)(3,11,12,8)(5,9,14,16)(6,10,13,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,2,5,8,6,10,12,9)(3,7,4,11,14,15,13,16)3668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 470);