metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊8D6, D40⋊6S3, D20⋊2D6, C24⋊11D10, D30.2D4, Dic6⋊2D10, D12.2D10, C120⋊15C22, Dic15.2D4, C60.142C23, C8⋊6(S3×D5), (C3×D40)⋊9C2, C24⋊C2⋊3D5, C5⋊2(D8⋊S3), C6.31(D4×D5), C40⋊S3⋊5C2, C15⋊D8⋊11C2, C3⋊2(D40⋊C2), C15⋊6(C8⋊C22), C10.31(S3×D4), C30.12(C2×D4), D20⋊S3⋊8C2, C20⋊D6⋊10C2, C15⋊3C8⋊20C22, C30.D4⋊10C2, C2.9(C20⋊D6), (C3×D20)⋊17C22, C20.71(C22×S3), C12.71(C22×D5), (C5×Dic6)⋊15C22, (C5×D12).26C22, (C4×D15).31C22, C4.115(C2×S3×D5), (C5×C24⋊C2)⋊5C2, SmallGroup(480,334)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊8D6
G = < a,b,c | a40=b6=c2=1, bab-1=a-1, cac=a29, cbc=b-1 >
Subgroups: 956 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C5×Dic3, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, D8⋊S3, C15⋊3C8, C120, D5×Dic3, C15⋊D4, C3⋊D20, C3×D20, C5×Dic6, C5×D12, C4×D15, C2×S3×D5, D40⋊C2, C15⋊D8, C30.D4, C3×D40, C5×C24⋊C2, C40⋊S3, D20⋊S3, C20⋊D6, C40⋊8D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D40⋊C2, C20⋊D6, C40⋊8D6
►On 120 points
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 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)
(1 106 51)(2 105 52 40 107 50)(3 104 53 39 108 49)(4 103 54 38 109 48)(5 102 55 37 110 47)(6 101 56 36 111 46)(7 100 57 35 112 45)(8 99 58 34 113 44)(9 98 59 33 114 43)(10 97 60 32 115 42)(11 96 61 31 116 41)(12 95 62 30 117 80)(13 94 63 29 118 79)(14 93 64 28 119 78)(15 92 65 27 120 77)(16 91 66 26 81 76)(17 90 67 25 82 75)(18 89 68 24 83 74)(19 88 69 23 84 73)(20 87 70 22 85 72)(21 86 71)
(1 51)(2 80)(3 69)(4 58)(5 47)(6 76)(7 65)(8 54)(9 43)(10 72)(11 61)(12 50)(13 79)(14 68)(15 57)(16 46)(17 75)(18 64)(19 53)(20 42)(21 71)(22 60)(23 49)(24 78)(25 67)(26 56)(27 45)(28 74)(29 63)(30 52)(31 41)(32 70)(33 59)(34 48)(35 77)(36 66)(37 55)(38 44)(39 73)(40 62)(81 101)(82 90)(83 119)(84 108)(85 97)(87 115)(88 104)(89 93)(91 111)(92 100)(94 118)(95 107)(98 114)(99 103)(102 110)(105 117)(109 113)(112 120)
G:=sub<Sym(120)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(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), (1,106,51)(2,105,52,40,107,50)(3,104,53,39,108,49)(4,103,54,38,109,48)(5,102,55,37,110,47)(6,101,56,36,111,46)(7,100,57,35,112,45)(8,99,58,34,113,44)(9,98,59,33,114,43)(10,97,60,32,115,42)(11,96,61,31,116,41)(12,95,62,30,117,80)(13,94,63,29,118,79)(14,93,64,28,119,78)(15,92,65,27,120,77)(16,91,66,26,81,76)(17,90,67,25,82,75)(18,89,68,24,83,74)(19,88,69,23,84,73)(20,87,70,22,85,72)(21,86,71), (1,51)(2,80)(3,69)(4,58)(5,47)(6,76)(7,65)(8,54)(9,43)(10,72)(11,61)(12,50)(13,79)(14,68)(15,57)(16,46)(17,75)(18,64)(19,53)(20,42)(21,71)(22,60)(23,49)(24,78)(25,67)(26,56)(27,45)(28,74)(29,63)(30,52)(31,41)(32,70)(33,59)(34,48)(35,77)(36,66)(37,55)(38,44)(39,73)(40,62)(81,101)(82,90)(83,119)(84,108)(85,97)(87,115)(88,104)(89,93)(91,111)(92,100)(94,118)(95,107)(98,114)(99,103)(102,110)(105,117)(109,113)(112,120)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(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), (1,106,51)(2,105,52,40,107,50)(3,104,53,39,108,49)(4,103,54,38,109,48)(5,102,55,37,110,47)(6,101,56,36,111,46)(7,100,57,35,112,45)(8,99,58,34,113,44)(9,98,59,33,114,43)(10,97,60,32,115,42)(11,96,61,31,116,41)(12,95,62,30,117,80)(13,94,63,29,118,79)(14,93,64,28,119,78)(15,92,65,27,120,77)(16,91,66,26,81,76)(17,90,67,25,82,75)(18,89,68,24,83,74)(19,88,69,23,84,73)(20,87,70,22,85,72)(21,86,71), (1,51)(2,80)(3,69)(4,58)(5,47)(6,76)(7,65)(8,54)(9,43)(10,72)(11,61)(12,50)(13,79)(14,68)(15,57)(16,46)(17,75)(18,64)(19,53)(20,42)(21,71)(22,60)(23,49)(24,78)(25,67)(26,56)(27,45)(28,74)(29,63)(30,52)(31,41)(32,70)(33,59)(34,48)(35,77)(36,66)(37,55)(38,44)(39,73)(40,62)(81,101)(82,90)(83,119)(84,108)(85,97)(87,115)(88,104)(89,93)(91,111)(92,100)(94,118)(95,107)(98,114)(99,103)(102,110)(105,117)(109,113)(112,120) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(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)], [(1,106,51),(2,105,52,40,107,50),(3,104,53,39,108,49),(4,103,54,38,109,48),(5,102,55,37,110,47),(6,101,56,36,111,46),(7,100,57,35,112,45),(8,99,58,34,113,44),(9,98,59,33,114,43),(10,97,60,32,115,42),(11,96,61,31,116,41),(12,95,62,30,117,80),(13,94,63,29,118,79),(14,93,64,28,119,78),(15,92,65,27,120,77),(16,91,66,26,81,76),(17,90,67,25,82,75),(18,89,68,24,83,74),(19,88,69,23,84,73),(20,87,70,22,85,72),(21,86,71)], [(1,51),(2,80),(3,69),(4,58),(5,47),(6,76),(7,65),(8,54),(9,43),(10,72),(11,61),(12,50),(13,79),(14,68),(15,57),(16,46),(17,75),(18,64),(19,53),(20,42),(21,71),(22,60),(23,49),(24,78),(25,67),(26,56),(27,45),(28,74),(29,63),(30,52),(31,41),(32,70),(33,59),(34,48),(35,77),(36,66),(37,55),(38,44),(39,73),(40,62),(81,101),(82,90),(83,119),(84,108),(85,97),(87,115),(88,104),(89,93),(91,111),(92,100),(94,118),(95,107),(98,114),(99,103),(102,110),(105,117),(109,113),(112,120)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 12 | 20 | 20 | 30 | 2 | 2 | 12 | 30 | 2 | 2 | 2 | 40 | 40 | 4 | 60 | 2 | 2 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | D8⋊S3 | C2×S3×D5 | D40⋊C2 | C20⋊D6 | C40⋊8D6 |
| kernel | C40⋊8D6 | C15⋊D8 | C30.D4 | C3×D40 | C5×C24⋊C2 | C40⋊S3 | D20⋊S3 | C20⋊D6 | D40 | Dic15 | D30 | C24⋊C2 | C40 | D20 | C24 | Dic6 | D12 | C15 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40⋊8D6 ►in GL8(𝔽241)
| 240 | 189 | 0 | 0 | 0 | 0 | 0 | 0 |
| 52 | 52 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 237 | 29 | 128 | 220 |
| 0 | 0 | 0 | 0 | 212 | 29 | 57 | 149 |
| 0 | 0 | 0 | 0 | 112 | 129 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 224 | 0 | 0 |
| 240 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 52 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 189 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 113 | 149 | 52 | 51 |
| 0 | 0 | 0 | 0 | 36 | 92 | 188 | 189 |
| 240 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 52 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 189 | 190 |
| 0 | 0 | 0 | 0 | 0 | 0 | 53 | 52 |
G:=sub<GL(8,GF(241))| [240,52,0,0,0,0,0,0,189,52,0,0,0,0,0,0,0,0,240,52,0,0,0,0,0,0,189,52,0,0,0,0,0,0,0,0,237,212,112,72,0,0,0,0,29,29,129,224,0,0,0,0,128,57,0,0,0,0,0,0,220,149,0,0],[240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,51,113,36,0,0,0,0,0,240,149,92,0,0,0,0,0,0,52,188,0,0,0,0,0,0,51,189],[240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,189,53,0,0,0,0,0,0,190,52] >;
C40⋊8D6 in GAP, Magma, Sage, TeX
C_{40}\rtimes_8D_6 % in TeX
G:=Group("C40:8D6"); // GroupNames label
G:=SmallGroup(480,334);
// by ID
G=gap.SmallGroup(480,334);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,142,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations