Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C24 |
| direct product | C131×#13075 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #1893 |
| commutator | C2×C8 | C22×C4×C131 |
| Frattini, cosocle | C22×C8 | C23×C131 |
| prime radical | C2×#41×C131 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C22×C4×C131 ◁ C2×#131×C131 ◁ G | |
458 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 195 points:
(1,2)(3,13)(4,19,21,9)(5,23)(6,28)(7,11)(8,16)(10,26)(12,25)(14,36)(15,37,44,57)(17,40)(18,50)(20,53,24,55)(22,34,54,52)(27,29)(30,43,51,46)(31,41)(32,48)(33,42,35,56)(38,49)(39,47)(45,63,62,59)(58,64,60,61)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(1,3,14,41,6,18,5,16,7,17,26,49,25,47,27,48)(2,8,29,50,12,31,10,13,11,32,23,39,28,38,36,40)(4,15,42,61,22,45,24,46,21,44,56,64,54,62,20,43)(9,30,55,63,34,58,35,37,19,51,53,59,52,60,33,57);(1,4)(2,9)(3,15)(5,24)(6,22)(7,21)(8,30)(10,35)(11,19)(12,34)(13,37)(14,42)(16,46)(17,44)(18,45)(20,27)(23,53)(25,54)(26,56)(28,52)(29,55)(31,58)(32,51)(33,36)(38,60)(39,59)(40,57)(41,61)(43,48)(47,62)(49,64)(50,63)6026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6063);