Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C131 · Schur multiplier (H2) = C4 |
| direct product | C131×#7534 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #1409 |
| commutator | C4×C8 | C2×C4×C131 |
| Frattini, cosocle | #494 | C22×C131 |
| prime radical | C22×C131 | #353 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C22×C131 ◁ C2×C4×C131 ◁ C42×C131 ◁ #494×C131 ◁ G | |
128 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 128 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) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 2 generators on 163 points:
(1,2,4,10,7,12,20,27)(3,9,14,25,16,23,30,32)(5,11,19,28,6,21,18,26)(8,17,22,13,24,29,31,15)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(2,8,12,24)(3,13)(5,6)(9,21,23,11)(10,25,27,32)(14,29)(15,16)(17,30)(18,19)(22,28,31,26)2489 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 522);