metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊2D4, C23.17D20, C5⋊6(C8⋊D4), C8⋊1(C5⋊D4), C40⋊5C4⋊18C2, (C2×C8).77D10, (C2×C4).51D20, C20.420(C2×D4), (C2×C20).297D4, (C2×M4(2))⋊1D5, D20⋊5C4⋊41C2, C20⋊7D4.17C2, (C10×M4(2))⋊1C2, (C2×C40).63C22, C20.230(C4○D4), C4.114(C4○D20), C20.44D4⋊41C2, C20.48D4⋊41C2, C2.22(C8⋊D10), C10.73(C4⋊D4), C2.21(C20⋊7D4), C10.22(C8⋊C22), (C2×C20).775C23, (C2×D20).22C22, (C22×C10).103D4, (C22×C4).142D10, C22.134(C2×D20), C4⋊Dic5.26C22, C2.22(C8.D10), C10.22(C8.C22), (C22×C20).304C22, (C2×Dic10).21C22, (C2×C40⋊C2)⋊2C2, C4.113(C2×C5⋊D4), (C2×C10).165(C2×D4), (C2×C4).724(C22×D5), SmallGroup(320,761)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊2D4
G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a19, cbc=b-1 >
Subgroups: 550 in 120 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, D5, C10 [×3], C10, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C40 [×2], C40, Dic10 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C8⋊D4, C40⋊C2 [×2], C10.D4, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C2×C40 [×2], C5×M4(2) [×2], C2×Dic10, C2×D20, C2×C5⋊D4, C22×C20, C20.44D4, C40⋊5C4, D20⋊5C4, C2×C40⋊C2, C20.48D4, C20⋊7D4, C10×M4(2), C40⋊2D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, D20 [×2], C5⋊D4 [×2], C22×D5, C8⋊D4, C2×D20, C4○D20, C2×C5⋊D4, C8⋊D10, C8.D10, C20⋊7D4, C40⋊2D4
►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)
(1 142 116 68)(2 141 117 67)(3 140 118 66)(4 139 119 65)(5 138 120 64)(6 137 81 63)(7 136 82 62)(8 135 83 61)(9 134 84 60)(10 133 85 59)(11 132 86 58)(12 131 87 57)(13 130 88 56)(14 129 89 55)(15 128 90 54)(16 127 91 53)(17 126 92 52)(18 125 93 51)(19 124 94 50)(20 123 95 49)(21 122 96 48)(22 121 97 47)(23 160 98 46)(24 159 99 45)(25 158 100 44)(26 157 101 43)(27 156 102 42)(28 155 103 41)(29 154 104 80)(30 153 105 79)(31 152 106 78)(32 151 107 77)(33 150 108 76)(34 149 109 75)(35 148 110 74)(36 147 111 73)(37 146 112 72)(38 145 113 71)(39 144 114 70)(40 143 115 69)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 149)(42 128)(43 147)(44 126)(45 145)(46 124)(47 143)(48 122)(49 141)(50 160)(51 139)(52 158)(53 137)(54 156)(55 135)(56 154)(57 133)(58 152)(59 131)(60 150)(61 129)(62 148)(63 127)(64 146)(65 125)(66 144)(67 123)(68 142)(69 121)(70 140)(71 159)(72 138)(73 157)(74 136)(75 155)(76 134)(77 153)(78 132)(79 151)(80 130)(81 91)(82 110)(83 89)(84 108)(85 87)(86 106)(88 104)(90 102)(92 100)(93 119)(94 98)(95 117)(97 115)(99 113)(101 111)(103 109)(105 107)(112 120)(114 118)
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), (1,142,116,68)(2,141,117,67)(3,140,118,66)(4,139,119,65)(5,138,120,64)(6,137,81,63)(7,136,82,62)(8,135,83,61)(9,134,84,60)(10,133,85,59)(11,132,86,58)(12,131,87,57)(13,130,88,56)(14,129,89,55)(15,128,90,54)(16,127,91,53)(17,126,92,52)(18,125,93,51)(19,124,94,50)(20,123,95,49)(21,122,96,48)(22,121,97,47)(23,160,98,46)(24,159,99,45)(25,158,100,44)(26,157,101,43)(27,156,102,42)(28,155,103,41)(29,154,104,80)(30,153,105,79)(31,152,106,78)(32,151,107,77)(33,150,108,76)(34,149,109,75)(35,148,110,74)(36,147,111,73)(37,146,112,72)(38,145,113,71)(39,144,114,70)(40,143,115,69), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,149)(42,128)(43,147)(44,126)(45,145)(46,124)(47,143)(48,122)(49,141)(50,160)(51,139)(52,158)(53,137)(54,156)(55,135)(56,154)(57,133)(58,152)(59,131)(60,150)(61,129)(62,148)(63,127)(64,146)(65,125)(66,144)(67,123)(68,142)(69,121)(70,140)(71,159)(72,138)(73,157)(74,136)(75,155)(76,134)(77,153)(78,132)(79,151)(80,130)(81,91)(82,110)(83,89)(84,108)(85,87)(86,106)(88,104)(90,102)(92,100)(93,119)(94,98)(95,117)(97,115)(99,113)(101,111)(103,109)(105,107)(112,120)(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)(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), (1,142,116,68)(2,141,117,67)(3,140,118,66)(4,139,119,65)(5,138,120,64)(6,137,81,63)(7,136,82,62)(8,135,83,61)(9,134,84,60)(10,133,85,59)(11,132,86,58)(12,131,87,57)(13,130,88,56)(14,129,89,55)(15,128,90,54)(16,127,91,53)(17,126,92,52)(18,125,93,51)(19,124,94,50)(20,123,95,49)(21,122,96,48)(22,121,97,47)(23,160,98,46)(24,159,99,45)(25,158,100,44)(26,157,101,43)(27,156,102,42)(28,155,103,41)(29,154,104,80)(30,153,105,79)(31,152,106,78)(32,151,107,77)(33,150,108,76)(34,149,109,75)(35,148,110,74)(36,147,111,73)(37,146,112,72)(38,145,113,71)(39,144,114,70)(40,143,115,69), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,149)(42,128)(43,147)(44,126)(45,145)(46,124)(47,143)(48,122)(49,141)(50,160)(51,139)(52,158)(53,137)(54,156)(55,135)(56,154)(57,133)(58,152)(59,131)(60,150)(61,129)(62,148)(63,127)(64,146)(65,125)(66,144)(67,123)(68,142)(69,121)(70,140)(71,159)(72,138)(73,157)(74,136)(75,155)(76,134)(77,153)(78,132)(79,151)(80,130)(81,91)(82,110)(83,89)(84,108)(85,87)(86,106)(88,104)(90,102)(92,100)(93,119)(94,98)(95,117)(97,115)(99,113)(101,111)(103,109)(105,107)(112,120)(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),(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)], [(1,142,116,68),(2,141,117,67),(3,140,118,66),(4,139,119,65),(5,138,120,64),(6,137,81,63),(7,136,82,62),(8,135,83,61),(9,134,84,60),(10,133,85,59),(11,132,86,58),(12,131,87,57),(13,130,88,56),(14,129,89,55),(15,128,90,54),(16,127,91,53),(17,126,92,52),(18,125,93,51),(19,124,94,50),(20,123,95,49),(21,122,96,48),(22,121,97,47),(23,160,98,46),(24,159,99,45),(25,158,100,44),(26,157,101,43),(27,156,102,42),(28,155,103,41),(29,154,104,80),(30,153,105,79),(31,152,106,78),(32,151,107,77),(33,150,108,76),(34,149,109,75),(35,148,110,74),(36,147,111,73),(37,146,112,72),(38,145,113,71),(39,144,114,70),(40,143,115,69)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,149),(42,128),(43,147),(44,126),(45,145),(46,124),(47,143),(48,122),(49,141),(50,160),(51,139),(52,158),(53,137),(54,156),(55,135),(56,154),(57,133),(58,152),(59,131),(60,150),(61,129),(62,148),(63,127),(64,146),(65,125),(66,144),(67,123),(68,142),(69,121),(70,140),(71,159),(72,138),(73,157),(74,136),(75,155),(76,134),(77,153),(78,132),(79,151),(80,130),(81,91),(82,110),(83,89),(84,108),(85,87),(86,106),(88,104),(90,102),(92,100),(93,119),(94,98),(95,117),(97,115),(99,113),(101,111),(103,109),(105,107),(112,120),(114,118)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 40 | 2 | 2 | 4 | 40 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | D20 | C4○D20 | C8⋊C22 | C8.C22 | C8⋊D10 | C8.D10 |
| kernel | C40⋊2D4 | C20.44D4 | C40⋊5C4 | D20⋊5C4 | C2×C40⋊C2 | C20.48D4 | C20⋊7D4 | C10×M4(2) | C40 | C2×C20 | C22×C10 | C2×M4(2) | C20 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C40⋊2D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 6 | 26 | 39 |
| 0 | 0 | 34 | 11 | 16 | 10 |
| 0 | 0 | 15 | 2 | 37 | 35 |
| 0 | 0 | 25 | 31 | 7 | 30 |
| 16 | 32 | 0 | 0 | 0 | 0 |
| 24 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 34 | 7 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,4,34,15,25,0,0,6,11,2,31,0,0,26,16,37,7,0,0,39,10,35,30],[16,24,0,0,0,0,32,25,0,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,34,34,0,0,0,0,1,7,0,0],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,1,7] >;
C40⋊2D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_2D_4 % in TeX
G:=Group("C40:2D4"); // GroupNames label
G:=SmallGroup(320,761);
// by ID
G=gap.SmallGroup(320,761);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations