metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊12D4, (C2×D8)⋊9D5, C8⋊4(C5⋊D4), C5⋊5(C8⋊2D4), (C10×D8)⋊10C2, C20⋊2D4⋊6C2, C40⋊6C4⋊21C2, (C2×C8).86D10, (C2×D4).66D10, C20.168(C2×D4), C20.94(C4○D4), D4⋊Dic5⋊31C2, (C2×Dic5).74D4, (C22×D5).44D4, C22.259(D4×D5), C4.29(D4⋊2D5), C2.31(D8⋊D5), C2.17(C20⋊2D4), C10.52(C8⋊C22), (C2×C40).148C22, (C2×C20).436C23, (D4×C10).85C22, C10.110(C4⋊D4), C4⋊Dic5.167C22, (C2×C8⋊D5)⋊7C2, C4.80(C2×C5⋊D4), (C2×C4×D5).50C22, (C2×C10).349(C2×D4), (C2×C4).526(C22×D5), (C2×C5⋊2C8).150C22, SmallGroup(320,786)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊12D4
G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a29, cbc=b-1 >
Subgroups: 566 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8, C2×C4, C2×C4 [×5], D4 [×8], C23 [×3], D5, C10, C10 [×2], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], D8 [×2], C22×C4, C2×D4 [×2], C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×6], D4⋊C4 [×2], C4.Q8, C4⋊D4 [×2], C2×M4(2), C2×D8, C5⋊2C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C8⋊2D4, C8⋊D5 [×2], C2×C5⋊2C8, C4⋊Dic5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10 [×2], C40⋊6C4, D4⋊Dic5 [×2], C2×C8⋊D5, C20⋊2D4 [×2], C10×D8, C40⋊12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C8⋊2D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D8⋊D5 [×2], C20⋊2D4, C40⋊12D4
►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 66 152 96)(2 45 153 115)(3 64 154 94)(4 43 155 113)(5 62 156 92)(6 41 157 111)(7 60 158 90)(8 79 159 109)(9 58 160 88)(10 77 121 107)(11 56 122 86)(12 75 123 105)(13 54 124 84)(14 73 125 103)(15 52 126 82)(16 71 127 101)(17 50 128 120)(18 69 129 99)(19 48 130 118)(20 67 131 97)(21 46 132 116)(22 65 133 95)(23 44 134 114)(24 63 135 93)(25 42 136 112)(26 61 137 91)(27 80 138 110)(28 59 139 89)(29 78 140 108)(30 57 141 87)(31 76 142 106)(32 55 143 85)(33 74 144 104)(34 53 145 83)(35 72 146 102)(36 51 147 81)(37 70 148 100)(38 49 149 119)(39 68 150 98)(40 47 151 117)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 91)(42 120)(43 109)(44 98)(45 87)(46 116)(47 105)(48 94)(49 83)(50 112)(51 101)(52 90)(53 119)(54 108)(55 97)(56 86)(57 115)(58 104)(59 93)(60 82)(61 111)(62 100)(63 89)(64 118)(65 107)(66 96)(67 85)(68 114)(69 103)(70 92)(71 81)(72 110)(73 99)(74 88)(75 117)(76 106)(77 95)(78 84)(79 113)(80 102)(121 133)(123 151)(124 140)(125 129)(126 158)(127 147)(128 136)(130 154)(131 143)(134 150)(135 139)(137 157)(138 146)(141 153)(144 160)(145 149)(148 156)(155 159)
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), (1,66,152,96)(2,45,153,115)(3,64,154,94)(4,43,155,113)(5,62,156,92)(6,41,157,111)(7,60,158,90)(8,79,159,109)(9,58,160,88)(10,77,121,107)(11,56,122,86)(12,75,123,105)(13,54,124,84)(14,73,125,103)(15,52,126,82)(16,71,127,101)(17,50,128,120)(18,69,129,99)(19,48,130,118)(20,67,131,97)(21,46,132,116)(22,65,133,95)(23,44,134,114)(24,63,135,93)(25,42,136,112)(26,61,137,91)(27,80,138,110)(28,59,139,89)(29,78,140,108)(30,57,141,87)(31,76,142,106)(32,55,143,85)(33,74,144,104)(34,53,145,83)(35,72,146,102)(36,51,147,81)(37,70,148,100)(38,49,149,119)(39,68,150,98)(40,47,151,117), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,91)(42,120)(43,109)(44,98)(45,87)(46,116)(47,105)(48,94)(49,83)(50,112)(51,101)(52,90)(53,119)(54,108)(55,97)(56,86)(57,115)(58,104)(59,93)(60,82)(61,111)(62,100)(63,89)(64,118)(65,107)(66,96)(67,85)(68,114)(69,103)(70,92)(71,81)(72,110)(73,99)(74,88)(75,117)(76,106)(77,95)(78,84)(79,113)(80,102)(121,133)(123,151)(124,140)(125,129)(126,158)(127,147)(128,136)(130,154)(131,143)(134,150)(135,139)(137,157)(138,146)(141,153)(144,160)(145,149)(148,156)(155,159)>;
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), (1,66,152,96)(2,45,153,115)(3,64,154,94)(4,43,155,113)(5,62,156,92)(6,41,157,111)(7,60,158,90)(8,79,159,109)(9,58,160,88)(10,77,121,107)(11,56,122,86)(12,75,123,105)(13,54,124,84)(14,73,125,103)(15,52,126,82)(16,71,127,101)(17,50,128,120)(18,69,129,99)(19,48,130,118)(20,67,131,97)(21,46,132,116)(22,65,133,95)(23,44,134,114)(24,63,135,93)(25,42,136,112)(26,61,137,91)(27,80,138,110)(28,59,139,89)(29,78,140,108)(30,57,141,87)(31,76,142,106)(32,55,143,85)(33,74,144,104)(34,53,145,83)(35,72,146,102)(36,51,147,81)(37,70,148,100)(38,49,149,119)(39,68,150,98)(40,47,151,117), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,91)(42,120)(43,109)(44,98)(45,87)(46,116)(47,105)(48,94)(49,83)(50,112)(51,101)(52,90)(53,119)(54,108)(55,97)(56,86)(57,115)(58,104)(59,93)(60,82)(61,111)(62,100)(63,89)(64,118)(65,107)(66,96)(67,85)(68,114)(69,103)(70,92)(71,81)(72,110)(73,99)(74,88)(75,117)(76,106)(77,95)(78,84)(79,113)(80,102)(121,133)(123,151)(124,140)(125,129)(126,158)(127,147)(128,136)(130,154)(131,143)(134,150)(135,139)(137,157)(138,146)(141,153)(144,160)(145,149)(148,156)(155,159) );
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)], [(1,66,152,96),(2,45,153,115),(3,64,154,94),(4,43,155,113),(5,62,156,92),(6,41,157,111),(7,60,158,90),(8,79,159,109),(9,58,160,88),(10,77,121,107),(11,56,122,86),(12,75,123,105),(13,54,124,84),(14,73,125,103),(15,52,126,82),(16,71,127,101),(17,50,128,120),(18,69,129,99),(19,48,130,118),(20,67,131,97),(21,46,132,116),(22,65,133,95),(23,44,134,114),(24,63,135,93),(25,42,136,112),(26,61,137,91),(27,80,138,110),(28,59,139,89),(29,78,140,108),(30,57,141,87),(31,76,142,106),(32,55,143,85),(33,74,144,104),(34,53,145,83),(35,72,146,102),(36,51,147,81),(37,70,148,100),(38,49,149,119),(39,68,150,98),(40,47,151,117)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,91),(42,120),(43,109),(44,98),(45,87),(46,116),(47,105),(48,94),(49,83),(50,112),(51,101),(52,90),(53,119),(54,108),(55,97),(56,86),(57,115),(58,104),(59,93),(60,82),(61,111),(62,100),(63,89),(64,118),(65,107),(66,96),(67,85),(68,114),(69,103),(70,92),(71,81),(72,110),(73,99),(74,88),(75,117),(76,106),(77,95),(78,84),(79,113),(80,102),(121,133),(123,151),(124,140),(125,129),(126,158),(127,147),(128,136),(130,154),(131,143),(134,150),(135,139),(137,157),(138,146),(141,153),(144,160),(145,149),(148,156),(155,159)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 20 | 2 | 2 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊D5 |
| kernel | C40⋊12D4 | C40⋊6C4 | D4⋊Dic5 | C2×C8⋊D5 | C20⋊2D4 | C10×D8 | C40 | C2×Dic5 | C22×D5 | C2×D8 | C20 | C2×C8 | C2×D4 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40⋊12D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 9 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 |
| 0 | 0 | 4 | 4 | 2 | 1 |
| 0 | 0 | 20 | 0 | 5 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 27 | 3 | 3 |
| 0 | 0 | 1 | 8 | 23 | 20 |
| 0 | 0 | 36 | 11 | 7 | 40 |
| 0 | 0 | 16 | 30 | 40 | 34 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 13 | 22 | 6 | 7 |
| 0 | 0 | 9 | 37 | 36 | 35 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,31,32,4,20,0,0,9,9,4,0,0,0,0,0,2,5,0,0,0,0,1,7],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,33,1,36,16,0,0,27,8,11,30,0,0,3,23,7,40,0,0,3,20,40,34],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,34,13,9,0,0,0,40,22,37,0,0,0,0,6,36,0,0,0,0,7,35] >;
C40⋊12D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{12}D_4 % in TeX
G:=Group("C40:12D4"); // GroupNames label
G:=SmallGroup(320,786);
// by ID
G=gap.SmallGroup(320,786);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations