metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊30D4, C23.24D20, C5⋊6(C8⋊8D4), (C22×C8)⋊9D5, C8⋊14(C5⋊D4), C40⋊6C4⋊18C2, (C2×C10)⋊8SD16, (C2×C4).67D20, D20⋊5C4⋊2C2, (C22×C40)⋊13C2, C20.412(C2×D4), (C2×C20).355D4, (C2×C8).320D10, C20⋊7D4.5C2, C10.17(C4○D8), C20.44D4⋊2C2, C20.48D4⋊2C2, C22⋊1(C40⋊C2), C10.17(C2×SD16), C4.111(C4○D20), C20.227(C4○D4), C2.18(C20⋊7D4), C10.70(C4⋊D4), (C2×C40).393C22, (C2×C20).768C23, (C2×D20).20C22, (C22×C10).140D4, C22.131(C2×D20), (C22×C4).430D10, C4⋊Dic5.23C22, C2.17(D40⋊7C2), (C22×C20).518C22, (C2×Dic10).19C22, (C2×C40⋊C2)⋊21C2, C2.17(C2×C40⋊C2), C4.105(C2×C5⋊D4), (C2×C10).158(C2×D4), (C2×C4).716(C22×D5), SmallGroup(320,741)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊30D4
G = < a,b,c | a40=b4=c2=1, bab-1=cac=a19, cbc=b-1 >
Subgroups: 550 in 124 conjugacy classes, 47 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, 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⋊8D4, C40⋊C2 [×2], C10.D4, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C2×C40 [×2], C2×C40 [×2], C2×Dic10, C2×D20, C2×C5⋊D4, C22×C20, C20.44D4, C40⋊6C4, D20⋊5C4, C2×C40⋊C2, C20.48D4, C20⋊7D4, C22×C40, C40⋊30D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, D20 [×2], C5⋊D4 [×2], C22×D5, C8⋊8D4, C40⋊C2 [×2], C2×D20, C4○D20, C2×C5⋊D4, C2×C40⋊C2, D40⋊7C2, C20⋊7D4, C40⋊30D4
►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)
(1 100 65 126)(2 119 66 145)(3 98 67 124)(4 117 68 143)(5 96 69 122)(6 115 70 141)(7 94 71 160)(8 113 72 139)(9 92 73 158)(10 111 74 137)(11 90 75 156)(12 109 76 135)(13 88 77 154)(14 107 78 133)(15 86 79 152)(16 105 80 131)(17 84 41 150)(18 103 42 129)(19 82 43 148)(20 101 44 127)(21 120 45 146)(22 99 46 125)(23 118 47 144)(24 97 48 123)(25 116 49 142)(26 95 50 121)(27 114 51 140)(28 93 52 159)(29 112 53 138)(30 91 54 157)(31 110 55 136)(32 89 56 155)(33 108 57 134)(34 87 58 153)(35 106 59 132)(36 85 60 151)(37 104 61 130)(38 83 62 149)(39 102 63 128)(40 81 64 147)
(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 49)(42 68)(43 47)(44 66)(46 64)(48 62)(50 60)(51 79)(52 58)(53 77)(54 56)(55 75)(57 73)(59 71)(61 69)(63 67)(70 80)(72 78)(74 76)(81 125)(82 144)(83 123)(84 142)(85 121)(86 140)(87 159)(88 138)(89 157)(90 136)(91 155)(92 134)(93 153)(94 132)(95 151)(96 130)(97 149)(98 128)(99 147)(100 126)(101 145)(102 124)(103 143)(104 122)(105 141)(106 160)(107 139)(108 158)(109 137)(110 156)(111 135)(112 154)(113 133)(114 152)(115 131)(116 150)(117 129)(118 148)(119 127)(120 146)
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), (1,100,65,126)(2,119,66,145)(3,98,67,124)(4,117,68,143)(5,96,69,122)(6,115,70,141)(7,94,71,160)(8,113,72,139)(9,92,73,158)(10,111,74,137)(11,90,75,156)(12,109,76,135)(13,88,77,154)(14,107,78,133)(15,86,79,152)(16,105,80,131)(17,84,41,150)(18,103,42,129)(19,82,43,148)(20,101,44,127)(21,120,45,146)(22,99,46,125)(23,118,47,144)(24,97,48,123)(25,116,49,142)(26,95,50,121)(27,114,51,140)(28,93,52,159)(29,112,53,138)(30,91,54,157)(31,110,55,136)(32,89,56,155)(33,108,57,134)(34,87,58,153)(35,106,59,132)(36,85,60,151)(37,104,61,130)(38,83,62,149)(39,102,63,128)(40,81,64,147), (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)(41,49)(42,68)(43,47)(44,66)(46,64)(48,62)(50,60)(51,79)(52,58)(53,77)(54,56)(55,75)(57,73)(59,71)(61,69)(63,67)(70,80)(72,78)(74,76)(81,125)(82,144)(83,123)(84,142)(85,121)(86,140)(87,159)(88,138)(89,157)(90,136)(91,155)(92,134)(93,153)(94,132)(95,151)(96,130)(97,149)(98,128)(99,147)(100,126)(101,145)(102,124)(103,143)(104,122)(105,141)(106,160)(107,139)(108,158)(109,137)(110,156)(111,135)(112,154)(113,133)(114,152)(115,131)(116,150)(117,129)(118,148)(119,127)(120,146)>;
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), (1,100,65,126)(2,119,66,145)(3,98,67,124)(4,117,68,143)(5,96,69,122)(6,115,70,141)(7,94,71,160)(8,113,72,139)(9,92,73,158)(10,111,74,137)(11,90,75,156)(12,109,76,135)(13,88,77,154)(14,107,78,133)(15,86,79,152)(16,105,80,131)(17,84,41,150)(18,103,42,129)(19,82,43,148)(20,101,44,127)(21,120,45,146)(22,99,46,125)(23,118,47,144)(24,97,48,123)(25,116,49,142)(26,95,50,121)(27,114,51,140)(28,93,52,159)(29,112,53,138)(30,91,54,157)(31,110,55,136)(32,89,56,155)(33,108,57,134)(34,87,58,153)(35,106,59,132)(36,85,60,151)(37,104,61,130)(38,83,62,149)(39,102,63,128)(40,81,64,147), (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)(41,49)(42,68)(43,47)(44,66)(46,64)(48,62)(50,60)(51,79)(52,58)(53,77)(54,56)(55,75)(57,73)(59,71)(61,69)(63,67)(70,80)(72,78)(74,76)(81,125)(82,144)(83,123)(84,142)(85,121)(86,140)(87,159)(88,138)(89,157)(90,136)(91,155)(92,134)(93,153)(94,132)(95,151)(96,130)(97,149)(98,128)(99,147)(100,126)(101,145)(102,124)(103,143)(104,122)(105,141)(106,160)(107,139)(108,158)(109,137)(110,156)(111,135)(112,154)(113,133)(114,152)(115,131)(116,150)(117,129)(118,148)(119,127)(120,146) );
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)], [(1,100,65,126),(2,119,66,145),(3,98,67,124),(4,117,68,143),(5,96,69,122),(6,115,70,141),(7,94,71,160),(8,113,72,139),(9,92,73,158),(10,111,74,137),(11,90,75,156),(12,109,76,135),(13,88,77,154),(14,107,78,133),(15,86,79,152),(16,105,80,131),(17,84,41,150),(18,103,42,129),(19,82,43,148),(20,101,44,127),(21,120,45,146),(22,99,46,125),(23,118,47,144),(24,97,48,123),(25,116,49,142),(26,95,50,121),(27,114,51,140),(28,93,52,159),(29,112,53,138),(30,91,54,157),(31,110,55,136),(32,89,56,155),(33,108,57,134),(34,87,58,153),(35,106,59,132),(36,85,60,151),(37,104,61,130),(38,83,62,149),(39,102,63,128),(40,81,64,147)], [(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),(41,49),(42,68),(43,47),(44,66),(46,64),(48,62),(50,60),(51,79),(52,58),(53,77),(54,56),(55,75),(57,73),(59,71),(61,69),(63,67),(70,80),(72,78),(74,76),(81,125),(82,144),(83,123),(84,142),(85,121),(86,140),(87,159),(88,138),(89,157),(90,136),(91,155),(92,134),(93,153),(94,132),(95,151),(96,130),(97,149),(98,128),(99,147),(100,126),(101,145),(102,124),(103,143),(104,122),(105,141),(106,160),(107,139),(108,158),(109,137),(110,156),(111,135),(112,154),(113,133),(114,152),(115,131),(116,150),(117,129),(118,148),(119,127),(120,146)])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 2 | 2 | 40 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | C4○D8 | C5⋊D4 | D20 | D20 | C4○D20 | C40⋊C2 | D40⋊7C2 |
| kernel | C40⋊30D4 | C20.44D4 | C40⋊6C4 | D20⋊5C4 | C2×C40⋊C2 | C20.48D4 | C20⋊7D4 | C22×C40 | C40 | C2×C20 | C22×C10 | C22×C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C40⋊30D4 ►in GL4(𝔽41) generated by
| 16 | 16 | 0 | 0 |
| 25 | 2 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 15 | 15 |
| 20 | 20 | 0 | 0 |
| 23 | 21 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 35 | 0 | 0 |
| 40 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [16,25,0,0,16,2,0,0,0,0,15,15,0,0,26,15],[20,23,0,0,20,21,0,0,0,0,40,0,0,0,0,1],[6,40,0,0,35,35,0,0,0,0,1,0,0,0,0,40] >;
C40⋊30D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{30}D_4 % in TeX
G:=Group("C40:30D4"); // GroupNames label
G:=SmallGroup(320,741);
// by ID
G=gap.SmallGroup(320,741);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=c*a*c=a^19,c*b*c=b^-1>;
// generators/relations