metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊11D4, (C2×D8)⋊8D5, C5⋊2C8⋊9D4, (C10×D8)⋊9C2, C8⋊3(C5⋊D4), C5⋊4(C8⋊3D4), C40⋊8C4⋊9C2, C4.21(D4×D5), C20⋊D4⋊5C2, (C2×C8).85D10, (C2×D4).62D10, C20.164(C2×D4), C20.17D4⋊4C2, (C2×Dic5).72D4, C22.254(D4×D5), C2.28(D8⋊D5), C2.18(C20⋊D4), C10.27(C4⋊1D4), C10.49(C8⋊C22), (C2×C20).431C23, (C2×C40).147C22, (D4×C10).81C22, (C2×D20).119C22, (C4×Dic5).51C22, (C2×Dic10).124C22, C4.5(C2×C5⋊D4), (C2×D4⋊D5)⋊18C2, (C2×C40⋊C2)⋊23C2, (C2×D4.D5)⋊17C2, (C2×C10).344(C2×D4), (C2×C4).521(C22×D5), (C2×C5⋊2C8).148C22, SmallGroup(320,781)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊11D4
G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a19, cbc=b-1 >
Subgroups: 686 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×3], D4 [×10], Q8 [×2], C23 [×3], D5, C10, C10 [×2], C10 [×2], C42, C22⋊C4 [×2], C2×C8, C2×C8, D8 [×4], SD16 [×4], C2×D4 [×2], C2×D4 [×3], C2×Q8, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×6], C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×D8, C2×SD16 [×2], C5⋊2C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C8⋊3D4, C40⋊C2 [×2], C2×C5⋊2C8, C4×Dic5, D4⋊D5 [×2], D4.D5 [×2], C23.D5 [×2], C2×C40, C5×D8 [×2], C2×Dic10, C2×D20, C2×C5⋊D4 [×2], D4×C10 [×2], C40⋊8C4, C2×C40⋊C2, C2×D4⋊D5, C2×D4.D5, C20.17D4, C20⋊D4, C10×D8, C40⋊11D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C4⋊1D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C8⋊3D4, D4×D5 [×2], C2×C5⋊D4, D8⋊D5 [×2], C20⋊D4, C40⋊11D4
►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)
(1 107 136 46)(2 96 137 75)(3 85 138 64)(4 114 139 53)(5 103 140 42)(6 92 141 71)(7 81 142 60)(8 110 143 49)(9 99 144 78)(10 88 145 67)(11 117 146 56)(12 106 147 45)(13 95 148 74)(14 84 149 63)(15 113 150 52)(16 102 151 41)(17 91 152 70)(18 120 153 59)(19 109 154 48)(20 98 155 77)(21 87 156 66)(22 116 157 55)(23 105 158 44)(24 94 159 73)(25 83 160 62)(26 112 121 51)(27 101 122 80)(28 90 123 69)(29 119 124 58)(30 108 125 47)(31 97 126 76)(32 86 127 65)(33 115 128 54)(34 104 129 43)(35 93 130 72)(36 82 131 61)(37 111 132 50)(38 100 133 79)(39 89 134 68)(40 118 135 57)
(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 92)(42 111)(43 90)(44 109)(45 88)(46 107)(47 86)(48 105)(49 84)(50 103)(51 82)(52 101)(53 120)(54 99)(55 118)(56 97)(57 116)(58 95)(59 114)(60 93)(61 112)(62 91)(63 110)(64 89)(65 108)(66 87)(67 106)(68 85)(69 104)(70 83)(71 102)(72 81)(73 100)(74 119)(75 98)(76 117)(77 96)(78 115)(79 94)(80 113)(121 131)(122 150)(123 129)(124 148)(125 127)(126 146)(128 144)(130 142)(132 140)(133 159)(134 138)(135 157)(137 155)(139 153)(141 151)(143 149)(145 147)(152 160)(154 158)
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), (1,107,136,46)(2,96,137,75)(3,85,138,64)(4,114,139,53)(5,103,140,42)(6,92,141,71)(7,81,142,60)(8,110,143,49)(9,99,144,78)(10,88,145,67)(11,117,146,56)(12,106,147,45)(13,95,148,74)(14,84,149,63)(15,113,150,52)(16,102,151,41)(17,91,152,70)(18,120,153,59)(19,109,154,48)(20,98,155,77)(21,87,156,66)(22,116,157,55)(23,105,158,44)(24,94,159,73)(25,83,160,62)(26,112,121,51)(27,101,122,80)(28,90,123,69)(29,119,124,58)(30,108,125,47)(31,97,126,76)(32,86,127,65)(33,115,128,54)(34,104,129,43)(35,93,130,72)(36,82,131,61)(37,111,132,50)(38,100,133,79)(39,89,134,68)(40,118,135,57), (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)(41,92)(42,111)(43,90)(44,109)(45,88)(46,107)(47,86)(48,105)(49,84)(50,103)(51,82)(52,101)(53,120)(54,99)(55,118)(56,97)(57,116)(58,95)(59,114)(60,93)(61,112)(62,91)(63,110)(64,89)(65,108)(66,87)(67,106)(68,85)(69,104)(70,83)(71,102)(72,81)(73,100)(74,119)(75,98)(76,117)(77,96)(78,115)(79,94)(80,113)(121,131)(122,150)(123,129)(124,148)(125,127)(126,146)(128,144)(130,142)(132,140)(133,159)(134,138)(135,157)(137,155)(139,153)(141,151)(143,149)(145,147)(152,160)(154,158)>;
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), (1,107,136,46)(2,96,137,75)(3,85,138,64)(4,114,139,53)(5,103,140,42)(6,92,141,71)(7,81,142,60)(8,110,143,49)(9,99,144,78)(10,88,145,67)(11,117,146,56)(12,106,147,45)(13,95,148,74)(14,84,149,63)(15,113,150,52)(16,102,151,41)(17,91,152,70)(18,120,153,59)(19,109,154,48)(20,98,155,77)(21,87,156,66)(22,116,157,55)(23,105,158,44)(24,94,159,73)(25,83,160,62)(26,112,121,51)(27,101,122,80)(28,90,123,69)(29,119,124,58)(30,108,125,47)(31,97,126,76)(32,86,127,65)(33,115,128,54)(34,104,129,43)(35,93,130,72)(36,82,131,61)(37,111,132,50)(38,100,133,79)(39,89,134,68)(40,118,135,57), (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)(41,92)(42,111)(43,90)(44,109)(45,88)(46,107)(47,86)(48,105)(49,84)(50,103)(51,82)(52,101)(53,120)(54,99)(55,118)(56,97)(57,116)(58,95)(59,114)(60,93)(61,112)(62,91)(63,110)(64,89)(65,108)(66,87)(67,106)(68,85)(69,104)(70,83)(71,102)(72,81)(73,100)(74,119)(75,98)(76,117)(77,96)(78,115)(79,94)(80,113)(121,131)(122,150)(123,129)(124,148)(125,127)(126,146)(128,144)(130,142)(132,140)(133,159)(134,138)(135,157)(137,155)(139,153)(141,151)(143,149)(145,147)(152,160)(154,158) );
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)], [(1,107,136,46),(2,96,137,75),(3,85,138,64),(4,114,139,53),(5,103,140,42),(6,92,141,71),(7,81,142,60),(8,110,143,49),(9,99,144,78),(10,88,145,67),(11,117,146,56),(12,106,147,45),(13,95,148,74),(14,84,149,63),(15,113,150,52),(16,102,151,41),(17,91,152,70),(18,120,153,59),(19,109,154,48),(20,98,155,77),(21,87,156,66),(22,116,157,55),(23,105,158,44),(24,94,159,73),(25,83,160,62),(26,112,121,51),(27,101,122,80),(28,90,123,69),(29,119,124,58),(30,108,125,47),(31,97,126,76),(32,86,127,65),(33,115,128,54),(34,104,129,43),(35,93,130,72),(36,82,131,61),(37,111,132,50),(38,100,133,79),(39,89,134,68),(40,118,135,57)], [(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),(41,92),(42,111),(43,90),(44,109),(45,88),(46,107),(47,86),(48,105),(49,84),(50,103),(51,82),(52,101),(53,120),(54,99),(55,118),(56,97),(57,116),(58,95),(59,114),(60,93),(61,112),(62,91),(63,110),(64,89),(65,108),(66,87),(67,106),(68,85),(69,104),(70,83),(71,102),(72,81),(73,100),(74,119),(75,98),(76,117),(77,96),(78,115),(79,94),(80,113),(121,131),(122,150),(123,129),(124,148),(125,127),(126,146),(128,144),(130,142),(132,140),(133,159),(134,138),(135,157),(137,155),(139,153),(141,151),(143,149),(145,147),(152,160),(154,158)])
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 | 40 | 2 | 2 | 20 | 20 | 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 | 1 | 1 | 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 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 |
| kernel | C40⋊11D4 | C40⋊8C4 | C2×C40⋊C2 | C2×D4⋊D5 | C2×D4.D5 | C20.17D4 | C20⋊D4 | C10×D8 | C5⋊2C8 | C40 | C2×Dic5 | C2×D8 | C2×C8 | C2×D4 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40⋊11D4 ►in GL10(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 38 | 20 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 35 | 20 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 37 | 21 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 15 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 38 | 11 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 31 | 11 | 24 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 36 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 13 | 18 | 34 |
| 17 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 24 | 20 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 20 | 26 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 24 | 21 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 21 | 21 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 11 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 25 | 33 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 37 | 32 | 0 | 34 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 4 | 39 | 15 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 23 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 7 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 17 | 40 | 40 |
G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20,26,21,21,0,0,0,0,0,0,38,35,37,0,0,0,0,0,0,0,20,20,21,15,0,0,0,0,0,0,4,0,3,6,0,0,0,0,0,0,0,0,0,0,35,31,3,6,0,0,0,0,0,0,38,11,36,13,0,0,0,0,0,0,11,24,2,18,0,0,0,0,0,0,11,7,0,34],[17,19,0,0,0,0,0,0,0,0,2,24,0,0,0,0,0,0,0,0,0,0,21,21,20,26,0,0,0,0,0,0,24,20,24,21,0,0,0,0,0,0,20,26,21,21,0,0,0,0,0,0,24,21,24,20,0,0,0,0,0,0,0,0,0,0,34,25,37,7,0,0,0,0,0,0,11,33,32,4,0,0,0,0,0,0,9,2,0,39,0,0,0,0,0,0,0,2,34,15],[40,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,1,23,22,26,0,0,0,0,0,0,0,40,7,17,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,40] >;
C40⋊11D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{11}D_4 % in TeX
G:=Group("C40:11D4"); // GroupNames label
G:=SmallGroup(320,781);
// by ID
G=gap.SmallGroup(320,781);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations