metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊29D4, C22⋊1D40, C23.25D20, (C2×C10)⋊5D8, (C2×D40)⋊9C2, C5⋊4(C8⋊7D4), (C22×C8)⋊6D5, C20⋊7D4⋊2C2, C8⋊13(C5⋊D4), C40⋊5C4⋊14C2, C10.17(C2×D8), (C2×C4).68D20, C2.17(C2×D40), D20⋊5C4⋊3C2, (C22×C40)⋊10C2, C20.413(C2×D4), (C2×C8).308D10, (C2×C20).356D4, C10.18(C4○D8), C4.112(C4○D20), C20.228(C4○D4), C10.71(C4⋊D4), C2.19(C20⋊7D4), (C2×C20).769C23, (C2×C40).380C22, (C2×D20).21C22, (C22×C10).141D4, C22.132(C2×D20), (C22×C4).431D10, C4⋊Dic5.24C22, C2.18(D40⋊7C2), (C22×C20).519C22, C4.106(C2×C5⋊D4), (C2×C10).159(C2×D4), (C2×C4).717(C22×D5), SmallGroup(320,742)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊29D4
G = < a,b,c | a40=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 694 in 134 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C40, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊7D4, D40, C4⋊Dic5, D10⋊C4, C2×C40, C2×C40, C2×D20, C2×C5⋊D4, C22×C20, C40⋊5C4, D20⋊5C4, C2×D40, C20⋊7D4, C22×C40, C40⋊29D4
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C4○D8, D20, C5⋊D4, C22×D5, C8⋊7D4, D40, C2×D20, C4○D20, C2×C5⋊D4, C2×D40, D40⋊7C2, C20⋊7D4, C40⋊29D4
►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)
(1 116 149 79)(2 115 150 78)(3 114 151 77)(4 113 152 76)(5 112 153 75)(6 111 154 74)(7 110 155 73)(8 109 156 72)(9 108 157 71)(10 107 158 70)(11 106 159 69)(12 105 160 68)(13 104 121 67)(14 103 122 66)(15 102 123 65)(16 101 124 64)(17 100 125 63)(18 99 126 62)(19 98 127 61)(20 97 128 60)(21 96 129 59)(22 95 130 58)(23 94 131 57)(24 93 132 56)(25 92 133 55)(26 91 134 54)(27 90 135 53)(28 89 136 52)(29 88 137 51)(30 87 138 50)(31 86 139 49)(32 85 140 48)(33 84 141 47)(34 83 142 46)(35 82 143 45)(36 81 144 44)(37 120 145 43)(38 119 146 42)(39 118 147 41)(40 117 148 80)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 120)(76 119)(77 118)(78 117)(79 116)(80 115)(121 137)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(138 160)(139 159)(140 158)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)
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), (1,116,149,79)(2,115,150,78)(3,114,151,77)(4,113,152,76)(5,112,153,75)(6,111,154,74)(7,110,155,73)(8,109,156,72)(9,108,157,71)(10,107,158,70)(11,106,159,69)(12,105,160,68)(13,104,121,67)(14,103,122,66)(15,102,123,65)(16,101,124,64)(17,100,125,63)(18,99,126,62)(19,98,127,61)(20,97,128,60)(21,96,129,59)(22,95,130,58)(23,94,131,57)(24,93,132,56)(25,92,133,55)(26,91,134,54)(27,90,135,53)(28,89,136,52)(29,88,137,51)(30,87,138,50)(31,86,139,49)(32,85,140,48)(33,84,141,47)(34,83,142,46)(35,82,143,45)(36,81,144,44)(37,120,145,43)(38,119,146,42)(39,118,147,41)(40,117,148,80), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,160)(139,159)(140,158)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)>;
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), (1,116,149,79)(2,115,150,78)(3,114,151,77)(4,113,152,76)(5,112,153,75)(6,111,154,74)(7,110,155,73)(8,109,156,72)(9,108,157,71)(10,107,158,70)(11,106,159,69)(12,105,160,68)(13,104,121,67)(14,103,122,66)(15,102,123,65)(16,101,124,64)(17,100,125,63)(18,99,126,62)(19,98,127,61)(20,97,128,60)(21,96,129,59)(22,95,130,58)(23,94,131,57)(24,93,132,56)(25,92,133,55)(26,91,134,54)(27,90,135,53)(28,89,136,52)(29,88,137,51)(30,87,138,50)(31,86,139,49)(32,85,140,48)(33,84,141,47)(34,83,142,46)(35,82,143,45)(36,81,144,44)(37,120,145,43)(38,119,146,42)(39,118,147,41)(40,117,148,80), (2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,160)(139,159)(140,158)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150) );
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)], [(1,116,149,79),(2,115,150,78),(3,114,151,77),(4,113,152,76),(5,112,153,75),(6,111,154,74),(7,110,155,73),(8,109,156,72),(9,108,157,71),(10,107,158,70),(11,106,159,69),(12,105,160,68),(13,104,121,67),(14,103,122,66),(15,102,123,65),(16,101,124,64),(17,100,125,63),(18,99,126,62),(19,98,127,61),(20,97,128,60),(21,96,129,59),(22,95,130,58),(23,94,131,57),(24,93,132,56),(25,92,133,55),(26,91,134,54),(27,90,135,53),(28,89,136,52),(29,88,137,51),(30,87,138,50),(31,86,139,49),(32,85,140,48),(33,84,141,47),(34,83,142,46),(35,82,143,45),(36,81,144,44),(37,120,145,43),(38,119,146,42),(39,118,147,41),(40,117,148,80)], [(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(41,114),(42,113),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83),(73,82),(74,81),(75,120),(76,119),(77,118),(78,117),(79,116),(80,115),(121,137),(122,136),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(138,160),(139,159),(140,158),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,150)]])
86 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 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 | D4 | D4 | D4 | D5 | C4○D4 | D8 | D10 | D10 | C4○D8 | C5⋊D4 | D20 | D20 | C4○D20 | D40 | D40⋊7C2 |
| kernel | C40⋊29D4 | C40⋊5C4 | D20⋊5C4 | C2×D40 | 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 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C40⋊29D4 ►in GL4(𝔽41) generated by
| 35 | 26 | 0 | 0 |
| 12 | 23 | 0 | 0 |
| 0 | 0 | 3 | 23 |
| 0 | 0 | 18 | 29 |
| 1 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 |
| 0 | 0 | 21 | 21 |
| 0 | 0 | 18 | 20 |
| 1 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 |
| 0 | 0 | 6 | 35 |
| 0 | 0 | 40 | 35 |
G:=sub<GL(4,GF(41))| [35,12,0,0,26,23,0,0,0,0,3,18,0,0,23,29],[1,9,0,0,0,40,0,0,0,0,21,18,0,0,21,20],[1,9,0,0,0,40,0,0,0,0,6,40,0,0,35,35] >;
C40⋊29D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{29}D_4 % in TeX
G:=Group("C40:29D4"); // GroupNames label
G:=SmallGroup(320,742);
// by ID
G=gap.SmallGroup(320,742);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations