Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C23 |
| direct product | C137×#13198 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C137 | #127 |
| commutator | C22×C4 | C22×C4×C137 |
| Frattini, cosocle | C22×C8 | C23×C137 |
| socle | C2×C137 | #1955 |
| prime radical | #575×C137 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C137 ◁ C22×C8×C137 ◁ G | |
410 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 201 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 201 points:
(1,2,5,10,6,11,21,32,7,12,22,33,23,34,47,58)(3,8,15,26,16,27,41,53,14,25,39,51,40,52,62,64)(4,9,19,30,20,31,36,49,18,29,43,55,46,57,61,59)(13,35,38,60,17,42,45,63,37,24,48,50,44,28,54,56)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201);(1,3,5,15,6,16,21,41,7,14,22,39,23,40,47,62)(2,8,10,26,11,27,32,53,12,25,33,51,34,52,58,64)(4,17,43,54,20,37,61,38,18,44,19,45,46,13,36,48)(9,28,55,63,31,35,59,50,29,42,30,56,57,24,49,60);(1,4)(2,9)(3,13,14,37)(5,19)(6,20)(7,18)(8,24,25,35)(10,30)(11,31)(12,29)(15,38,39,48)(16,17,40,44)(21,36)(22,43)(23,46)(26,50,51,60)(27,28,52,42)(32,49)(33,55)(34,57)(41,45,62,54)(47,61)(53,56,64,63)(58,59)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 6186);