metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊15C4, C8⋊4(C4×D5), C40⋊18(C2×C4), C4.Q8⋊5D5, C40⋊8C4⋊3C2, D20⋊16(C2×C4), C5⋊4(D8⋊C4), C10.81(C4×D4), (C2×C8).63D10, D20⋊8C4⋊6C2, C4⋊C4.163D10, (C2×D40).13C2, D20⋊6C4⋊18C2, C22.86(D4×D5), C20.34(C4○D4), C2.6(D40⋊C2), C4.6(Q8⋊2D5), C10.72(C8⋊C22), C20.104(C22×C4), (C2×C20).285C23, (C2×C40).112C22, (C2×Dic5).221D4, (C2×D20).81C22, C2.11(D20⋊8C4), (C4×Dic5).37C22, C4.42(C2×C4×D5), (C5×C4.Q8)⋊5C2, (C2×C10).290(C2×D4), (C5×C4⋊C4).78C22, (C2×C5⋊2C8).62C22, (C2×C4).388(C22×D5), SmallGroup(320,496)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊15C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >
Subgroups: 646 in 132 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, D8⋊C4, D40, C2×C5⋊2C8, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, D20⋊6C4, C40⋊8C4, C5×C4.Q8, D20⋊8C4, C2×D40, D40⋊15C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C8⋊C22, C4×D5, C22×D5, D8⋊C4, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, D40⋊C2, D40⋊15C4
►On 160 points
Generators in S160
(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)(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(121 129)(122 128)(123 127)(124 126)(130 160)(131 159)(132 158)(133 157)(134 156)(135 155)(136 154)(137 153)(138 152)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)
(1 87 138 46)(2 98 139 57)(3 109 140 68)(4 120 141 79)(5 91 142 50)(6 102 143 61)(7 113 144 72)(8 84 145 43)(9 95 146 54)(10 106 147 65)(11 117 148 76)(12 88 149 47)(13 99 150 58)(14 110 151 69)(15 81 152 80)(16 92 153 51)(17 103 154 62)(18 114 155 73)(19 85 156 44)(20 96 157 55)(21 107 158 66)(22 118 159 77)(23 89 160 48)(24 100 121 59)(25 111 122 70)(26 82 123 41)(27 93 124 52)(28 104 125 63)(29 115 126 74)(30 86 127 45)(31 97 128 56)(32 108 129 67)(33 119 130 78)(34 90 131 49)(35 101 132 60)(36 112 133 71)(37 83 134 42)(38 94 135 53)(39 105 136 64)(40 116 137 75)
G:=sub<Sym(160)| (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)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(121,129)(122,128)(123,127)(124,126)(130,160)(131,159)(132,158)(133,157)(134,156)(135,155)(136,154)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146), (1,87,138,46)(2,98,139,57)(3,109,140,68)(4,120,141,79)(5,91,142,50)(6,102,143,61)(7,113,144,72)(8,84,145,43)(9,95,146,54)(10,106,147,65)(11,117,148,76)(12,88,149,47)(13,99,150,58)(14,110,151,69)(15,81,152,80)(16,92,153,51)(17,103,154,62)(18,114,155,73)(19,85,156,44)(20,96,157,55)(21,107,158,66)(22,118,159,77)(23,89,160,48)(24,100,121,59)(25,111,122,70)(26,82,123,41)(27,93,124,52)(28,104,125,63)(29,115,126,74)(30,86,127,45)(31,97,128,56)(32,108,129,67)(33,119,130,78)(34,90,131,49)(35,101,132,60)(36,112,133,71)(37,83,134,42)(38,94,135,53)(39,105,136,64)(40,116,137,75)>;
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)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,90)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(121,129)(122,128)(123,127)(124,126)(130,160)(131,159)(132,158)(133,157)(134,156)(135,155)(136,154)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146), (1,87,138,46)(2,98,139,57)(3,109,140,68)(4,120,141,79)(5,91,142,50)(6,102,143,61)(7,113,144,72)(8,84,145,43)(9,95,146,54)(10,106,147,65)(11,117,148,76)(12,88,149,47)(13,99,150,58)(14,110,151,69)(15,81,152,80)(16,92,153,51)(17,103,154,62)(18,114,155,73)(19,85,156,44)(20,96,157,55)(21,107,158,66)(22,118,159,77)(23,89,160,48)(24,100,121,59)(25,111,122,70)(26,82,123,41)(27,93,124,52)(28,104,125,63)(29,115,126,74)(30,86,127,45)(31,97,128,56)(32,108,129,67)(33,119,130,78)(34,90,131,49)(35,101,132,60)(36,112,133,71)(37,83,134,42)(38,94,135,53)(39,105,136,64)(40,116,137,75) );
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),(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,80),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,90),(98,120),(99,119),(100,118),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(121,129),(122,128),(123,127),(124,126),(130,160),(131,159),(132,158),(133,157),(134,156),(135,155),(136,154),(137,153),(138,152),(139,151),(140,150),(141,149),(142,148),(143,147),(144,146)], [(1,87,138,46),(2,98,139,57),(3,109,140,68),(4,120,141,79),(5,91,142,50),(6,102,143,61),(7,113,144,72),(8,84,145,43),(9,95,146,54),(10,106,147,65),(11,117,148,76),(12,88,149,47),(13,99,150,58),(14,110,151,69),(15,81,152,80),(16,92,153,51),(17,103,154,62),(18,114,155,73),(19,85,156,44),(20,96,157,55),(21,107,158,66),(22,118,159,77),(23,89,160,48),(24,100,121,59),(25,111,122,70),(26,82,123,41),(27,93,124,52),(28,104,125,63),(29,115,126,74),(30,86,127,45),(31,97,128,56),(32,108,129,67),(33,119,130,78),(34,90,131,49),(35,101,132,60),(36,112,133,71),(37,83,134,42),(38,94,135,53),(39,105,136,64),(40,116,137,75)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | C8⋊C22 | Q8⋊2D5 | D4×D5 | D40⋊C2 |
| kernel | D40⋊15C4 | D20⋊6C4 | C40⋊8C4 | C5×C4.Q8 | D20⋊8C4 | C2×D40 | D40 | C2×Dic5 | C4.Q8 | C20 | C4⋊C4 | C2×C8 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 4 | 2 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of D40⋊15C4 ►in GL6(𝔽41)
| 30 | 36 | 0 | 0 | 0 | 0 |
| 8 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 28 | 13 | 28 |
| 0 | 0 | 13 | 17 | 13 | 17 |
| 0 | 0 | 28 | 13 | 13 | 28 |
| 0 | 0 | 28 | 24 | 13 | 17 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 29 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 1 | 35 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 24 | 37 | 0 | 0 | 0 | 0 |
| 11 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 24 | 18 | 17 |
| 0 | 0 | 17 | 18 | 24 | 38 |
| 0 | 0 | 18 | 17 | 3 | 17 |
| 0 | 0 | 24 | 38 | 24 | 23 |
G:=sub<GL(6,GF(41))| [30,8,0,0,0,0,36,11,0,0,0,0,0,0,13,13,28,28,0,0,28,17,13,24,0,0,13,13,13,13,0,0,28,17,28,17],[40,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,6,1,0,0,0,0,6,35,0,0],[24,11,0,0,0,0,37,17,0,0,0,0,0,0,38,17,18,24,0,0,24,18,17,38,0,0,18,24,3,24,0,0,17,38,17,23] >;
D40⋊15C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_{15}C_4 % in TeX
G:=Group("D40:15C4"); // GroupNames label
G:=SmallGroup(320,496);
// by ID
G=gap.SmallGroup(320,496);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,219,58,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations