Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C131 · Schur multiplier (H2) = C22 |
| direct product | C131×#7395 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #1343 |
| commutator | C2×C22⋊C4 | C2×C4×C131 |
| Frattini, cosocle | #381 | C22×C131 |
| prime radical | #534×C131 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C23×C131 ◁ C24×C131 ◁ #381×C131 ◁ G | |
284 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 2 generators on 163 points:
(1,2,5,9,6,10,18,26)(3,11,13,27,14,23,30,32)(4,15,16,31,17,8,22,24)(7,19,20,12,21,28,25,29)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,3)(2,7)(4,12)(5,13)(6,14)(8,23)(9,25)(10,21)(11,15)(16,19)(17,29)(18,30)(20,26)(22,28)(24,32)(27,31)3275 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 383);