metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊5D6, D40⋊4S3, D15⋊1D8, D20⋊1D6, D24⋊4D5, C24⋊5D10, D12⋊1D10, D30.21D4, C120⋊14C22, C60.140C23, Dic15.26D4, C5⋊2(S3×D8), C3⋊2(D5×D8), C15⋊3(C2×D8), C8⋊13(S3×D5), (C3×D40)⋊7C2, (C8×D15)⋊6C2, (C5×D24)⋊6C2, C15⋊D8⋊9C2, C6.29(D4×D5), C20⋊D6⋊8C2, C30.10(C2×D4), C10.29(S3×D4), C15⋊3C8⋊36C22, C2.7(C20⋊D6), (C5×D12)⋊16C22, (C3×D20)⋊16C22, C20.69(C22×S3), C12.69(C22×D5), (C4×D15).54C22, C4.113(C2×S3×D5), SmallGroup(480,332)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊5D6
G = < a,b,c | a40=b6=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >
Subgroups: 1180 in 152 conjugacy classes, 40 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, C2×D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C2×D8, C5⋊2C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, S3×D8, C15⋊3C8, C120, C15⋊D4, C3×D20, C5×D12, C4×D15, C2×S3×D5, D5×D8, C15⋊D8, C3×D40, C5×D24, C8×D15, C20⋊D6, C40⋊5D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, C22×S3, C2×D8, C22×D5, S3×D4, S3×D5, D4×D5, S3×D8, C2×S3×D5, D5×D8, C20⋊D6, C40⋊5D6
►On 120 points
Generators in S120
(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 81 69 16 106 44)(2 120 70 15 107 43)(3 119 71 14 108 42)(4 118 72 13 109 41)(5 117 73 12 110 80)(6 116 74 11 111 79)(7 115 75 10 112 78)(8 114 76 9 113 77)(17 105 45 40 82 68)(18 104 46 39 83 67)(19 103 47 38 84 66)(20 102 48 37 85 65)(21 101 49 36 86 64)(22 100 50 35 87 63)(23 99 51 34 88 62)(24 98 52 33 89 61)(25 97 53 32 90 60)(26 96 54 31 91 59)(27 95 55 30 92 58)(28 94 56 29 93 57)
(1 49)(2 58)(3 67)(4 76)(5 45)(6 54)(7 63)(8 72)(9 41)(10 50)(11 59)(12 68)(13 77)(14 46)(15 55)(16 64)(17 73)(18 42)(19 51)(20 60)(21 69)(22 78)(23 47)(24 56)(25 65)(26 74)(27 43)(28 52)(29 61)(30 70)(31 79)(32 48)(33 57)(34 66)(35 75)(36 44)(37 53)(38 62)(39 71)(40 80)(81 101)(82 110)(83 119)(84 88)(85 97)(86 106)(87 115)(89 93)(90 102)(91 111)(92 120)(94 98)(95 107)(96 116)(99 103)(100 112)(104 108)(105 117)(109 113)(114 118)
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), (1,81,69,16,106,44)(2,120,70,15,107,43)(3,119,71,14,108,42)(4,118,72,13,109,41)(5,117,73,12,110,80)(6,116,74,11,111,79)(7,115,75,10,112,78)(8,114,76,9,113,77)(17,105,45,40,82,68)(18,104,46,39,83,67)(19,103,47,38,84,66)(20,102,48,37,85,65)(21,101,49,36,86,64)(22,100,50,35,87,63)(23,99,51,34,88,62)(24,98,52,33,89,61)(25,97,53,32,90,60)(26,96,54,31,91,59)(27,95,55,30,92,58)(28,94,56,29,93,57), (1,49)(2,58)(3,67)(4,76)(5,45)(6,54)(7,63)(8,72)(9,41)(10,50)(11,59)(12,68)(13,77)(14,46)(15,55)(16,64)(17,73)(18,42)(19,51)(20,60)(21,69)(22,78)(23,47)(24,56)(25,65)(26,74)(27,43)(28,52)(29,61)(30,70)(31,79)(32,48)(33,57)(34,66)(35,75)(36,44)(37,53)(38,62)(39,71)(40,80)(81,101)(82,110)(83,119)(84,88)(85,97)(86,106)(87,115)(89,93)(90,102)(91,111)(92,120)(94,98)(95,107)(96,116)(99,103)(100,112)(104,108)(105,117)(109,113)(114,118)>;
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), (1,81,69,16,106,44)(2,120,70,15,107,43)(3,119,71,14,108,42)(4,118,72,13,109,41)(5,117,73,12,110,80)(6,116,74,11,111,79)(7,115,75,10,112,78)(8,114,76,9,113,77)(17,105,45,40,82,68)(18,104,46,39,83,67)(19,103,47,38,84,66)(20,102,48,37,85,65)(21,101,49,36,86,64)(22,100,50,35,87,63)(23,99,51,34,88,62)(24,98,52,33,89,61)(25,97,53,32,90,60)(26,96,54,31,91,59)(27,95,55,30,92,58)(28,94,56,29,93,57), (1,49)(2,58)(3,67)(4,76)(5,45)(6,54)(7,63)(8,72)(9,41)(10,50)(11,59)(12,68)(13,77)(14,46)(15,55)(16,64)(17,73)(18,42)(19,51)(20,60)(21,69)(22,78)(23,47)(24,56)(25,65)(26,74)(27,43)(28,52)(29,61)(30,70)(31,79)(32,48)(33,57)(34,66)(35,75)(36,44)(37,53)(38,62)(39,71)(40,80)(81,101)(82,110)(83,119)(84,88)(85,97)(86,106)(87,115)(89,93)(90,102)(91,111)(92,120)(94,98)(95,107)(96,116)(99,103)(100,112)(104,108)(105,117)(109,113)(114,118) );
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)], [(1,81,69,16,106,44),(2,120,70,15,107,43),(3,119,71,14,108,42),(4,118,72,13,109,41),(5,117,73,12,110,80),(6,116,74,11,111,79),(7,115,75,10,112,78),(8,114,76,9,113,77),(17,105,45,40,82,68),(18,104,46,39,83,67),(19,103,47,38,84,66),(20,102,48,37,85,65),(21,101,49,36,86,64),(22,100,50,35,87,63),(23,99,51,34,88,62),(24,98,52,33,89,61),(25,97,53,32,90,60),(26,96,54,31,91,59),(27,95,55,30,92,58),(28,94,56,29,93,57)], [(1,49),(2,58),(3,67),(4,76),(5,45),(6,54),(7,63),(8,72),(9,41),(10,50),(11,59),(12,68),(13,77),(14,46),(15,55),(16,64),(17,73),(18,42),(19,51),(20,60),(21,69),(22,78),(23,47),(24,56),(25,65),(26,74),(27,43),(28,52),(29,61),(30,70),(31,79),(32,48),(33,57),(34,66),(35,75),(36,44),(37,53),(38,62),(39,71),(40,80),(81,101),(82,110),(83,119),(84,88),(85,97),(86,106),(87,115),(89,93),(90,102),(91,111),(92,120),(94,98),(95,107),(96,116),(99,103),(100,112),(104,108),(105,117),(109,113),(114,118)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 12 | 12 | 15 | 15 | 20 | 20 | 2 | 2 | 30 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 30 | 30 | 2 | 2 | 24 | 24 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D8 | D10 | D10 | S3×D4 | S3×D5 | D4×D5 | S3×D8 | C2×S3×D5 | D5×D8 | C20⋊D6 | C40⋊5D6 |
| kernel | C40⋊5D6 | C15⋊D8 | C3×D40 | C5×D24 | C8×D15 | C20⋊D6 | D40 | Dic15 | D30 | D24 | C40 | D20 | D15 | C24 | D12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40⋊5D6 ►in GL6(𝔽241)
| 0 | 190 | 0 | 0 | 0 | 0 |
| 52 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 230 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 52 | 51 | 0 | 0 | 0 | 0 |
| 188 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 189 | 190 | 0 | 0 | 0 | 0 |
| 53 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [0,52,0,0,0,0,190,52,0,0,0,0,0,0,0,230,0,0,0,0,22,22,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[52,188,0,0,0,0,51,189,0,0,0,0,0,0,0,11,0,0,0,0,22,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[189,53,0,0,0,0,190,52,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;
C40⋊5D6 in GAP, Magma, Sage, TeX
C_{40}\rtimes_5D_6 % in TeX
G:=Group("C40:5D6"); // GroupNames label
G:=SmallGroup(480,332);
// by ID
G=gap.SmallGroup(480,332);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,142,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations