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, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C40, C40, Dic10, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊8D4, C40⋊C2, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C40, C2×C40, 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, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, D20, C5⋊D4, C22×D5, C8⋊8D4, C40⋊C2, 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 42 87 139)(2 61 88 158)(3 80 89 137)(4 59 90 156)(5 78 91 135)(6 57 92 154)(7 76 93 133)(8 55 94 152)(9 74 95 131)(10 53 96 150)(11 72 97 129)(12 51 98 148)(13 70 99 127)(14 49 100 146)(15 68 101 125)(16 47 102 144)(17 66 103 123)(18 45 104 142)(19 64 105 121)(20 43 106 140)(21 62 107 159)(22 41 108 138)(23 60 109 157)(24 79 110 136)(25 58 111 155)(26 77 112 134)(27 56 113 153)(28 75 114 132)(29 54 115 151)(30 73 116 130)(31 52 117 149)(32 71 118 128)(33 50 119 147)(34 69 120 126)(35 48 81 145)(36 67 82 124)(37 46 83 143)(38 65 84 122)(39 44 85 141)(40 63 86 160)
(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)(41 160)(42 139)(43 158)(44 137)(45 156)(46 135)(47 154)(48 133)(49 152)(50 131)(51 150)(52 129)(53 148)(54 127)(55 146)(56 125)(57 144)(58 123)(59 142)(60 121)(61 140)(62 159)(63 138)(64 157)(65 136)(66 155)(67 134)(68 153)(69 132)(70 151)(71 130)(72 149)(73 128)(74 147)(75 126)(76 145)(77 124)(78 143)(79 122)(80 141)(81 93)(82 112)(83 91)(84 110)(85 89)(86 108)(88 106)(90 104)(92 102)(94 100)(95 119)(96 98)(97 117)(99 115)(101 113)(103 111)(105 109)(114 120)(116 118)
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), (1,42,87,139)(2,61,88,158)(3,80,89,137)(4,59,90,156)(5,78,91,135)(6,57,92,154)(7,76,93,133)(8,55,94,152)(9,74,95,131)(10,53,96,150)(11,72,97,129)(12,51,98,148)(13,70,99,127)(14,49,100,146)(15,68,101,125)(16,47,102,144)(17,66,103,123)(18,45,104,142)(19,64,105,121)(20,43,106,140)(21,62,107,159)(22,41,108,138)(23,60,109,157)(24,79,110,136)(25,58,111,155)(26,77,112,134)(27,56,113,153)(28,75,114,132)(29,54,115,151)(30,73,116,130)(31,52,117,149)(32,71,118,128)(33,50,119,147)(34,69,120,126)(35,48,81,145)(36,67,82,124)(37,46,83,143)(38,65,84,122)(39,44,85,141)(40,63,86,160), (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)(41,160)(42,139)(43,158)(44,137)(45,156)(46,135)(47,154)(48,133)(49,152)(50,131)(51,150)(52,129)(53,148)(54,127)(55,146)(56,125)(57,144)(58,123)(59,142)(60,121)(61,140)(62,159)(63,138)(64,157)(65,136)(66,155)(67,134)(68,153)(69,132)(70,151)(71,130)(72,149)(73,128)(74,147)(75,126)(76,145)(77,124)(78,143)(79,122)(80,141)(81,93)(82,112)(83,91)(84,110)(85,89)(86,108)(88,106)(90,104)(92,102)(94,100)(95,119)(96,98)(97,117)(99,115)(101,113)(103,111)(105,109)(114,120)(116,118)>;
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), (1,42,87,139)(2,61,88,158)(3,80,89,137)(4,59,90,156)(5,78,91,135)(6,57,92,154)(7,76,93,133)(8,55,94,152)(9,74,95,131)(10,53,96,150)(11,72,97,129)(12,51,98,148)(13,70,99,127)(14,49,100,146)(15,68,101,125)(16,47,102,144)(17,66,103,123)(18,45,104,142)(19,64,105,121)(20,43,106,140)(21,62,107,159)(22,41,108,138)(23,60,109,157)(24,79,110,136)(25,58,111,155)(26,77,112,134)(27,56,113,153)(28,75,114,132)(29,54,115,151)(30,73,116,130)(31,52,117,149)(32,71,118,128)(33,50,119,147)(34,69,120,126)(35,48,81,145)(36,67,82,124)(37,46,83,143)(38,65,84,122)(39,44,85,141)(40,63,86,160), (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)(41,160)(42,139)(43,158)(44,137)(45,156)(46,135)(47,154)(48,133)(49,152)(50,131)(51,150)(52,129)(53,148)(54,127)(55,146)(56,125)(57,144)(58,123)(59,142)(60,121)(61,140)(62,159)(63,138)(64,157)(65,136)(66,155)(67,134)(68,153)(69,132)(70,151)(71,130)(72,149)(73,128)(74,147)(75,126)(76,145)(77,124)(78,143)(79,122)(80,141)(81,93)(82,112)(83,91)(84,110)(85,89)(86,108)(88,106)(90,104)(92,102)(94,100)(95,119)(96,98)(97,117)(99,115)(101,113)(103,111)(105,109)(114,120)(116,118) );
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)], [(1,42,87,139),(2,61,88,158),(3,80,89,137),(4,59,90,156),(5,78,91,135),(6,57,92,154),(7,76,93,133),(8,55,94,152),(9,74,95,131),(10,53,96,150),(11,72,97,129),(12,51,98,148),(13,70,99,127),(14,49,100,146),(15,68,101,125),(16,47,102,144),(17,66,103,123),(18,45,104,142),(19,64,105,121),(20,43,106,140),(21,62,107,159),(22,41,108,138),(23,60,109,157),(24,79,110,136),(25,58,111,155),(26,77,112,134),(27,56,113,153),(28,75,114,132),(29,54,115,151),(30,73,116,130),(31,52,117,149),(32,71,118,128),(33,50,119,147),(34,69,120,126),(35,48,81,145),(36,67,82,124),(37,46,83,143),(38,65,84,122),(39,44,85,141),(40,63,86,160)], [(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),(41,160),(42,139),(43,158),(44,137),(45,156),(46,135),(47,154),(48,133),(49,152),(50,131),(51,150),(52,129),(53,148),(54,127),(55,146),(56,125),(57,144),(58,123),(59,142),(60,121),(61,140),(62,159),(63,138),(64,157),(65,136),(66,155),(67,134),(68,153),(69,132),(70,151),(71,130),(72,149),(73,128),(74,147),(75,126),(76,145),(77,124),(78,143),(79,122),(80,141),(81,93),(82,112),(83,91),(84,110),(85,89),(86,108),(88,106),(90,104),(92,102),(94,100),(95,119),(96,98),(97,117),(99,115),(101,113),(103,111),(105,109),(114,120),(116,118)]])
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