Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C22 |
| direct product | C137×#11914 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #1530 |
| commutator | C42 | C22×C4×C137 |
| Frattini, cosocle | #97 | C23×C137 |
| prime radical | C23×C137 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C2×C4×C137 ◁ #405×C137 ◁ G | |
272 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,5,11,6,12,26,42,7,13,25,43,28,10,37,39)(3,14,16,45,17,8,29,31,18,32,50,40,27,33,23,56)(4,19,21,51,22,36,54,41,20,38,24,58,52,9,34,35)(15,30,47,60,48,59,57,63,46,61,49,62,55,44,53,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201);(1,3,7,18)(2,8,13,33)(4,15,20,46)(5,23,25,29)(6,27,28,17)(9,30,36,61)(10,32,12,14)(11,40,43,45)(16,26,50,37)(19,44,38,59)(21,53,24,57)(22,55,52,48)(31,39,56,42)(34,47,54,49)(35,63,41,64)(51,60,58,62);(1,4,7,20)(2,9)(3,15,18,46)(5,24,25,21)(6,22,28,52)(8,30)(10,38)(11,41)(12,19)(13,36)(14,44)(16,49,50,47)(17,48,27,55)(23,57,29,53)(26,34,37,54)(31,62)(32,59)(33,61)(35,43)(39,51)(40,64)(42,58)(45,63)(56,60)4658 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 4902);