metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊2D4, C23.17D20, C5⋊6(C8⋊D4), C8⋊1(C5⋊D4), C40⋊5C4⋊18C2, (C2×C8).77D10, (C2×C4).51D20, C20.420(C2×D4), (C2×C20).297D4, (C2×M4(2))⋊1D5, D20⋊5C4⋊41C2, C20⋊7D4.17C2, (C10×M4(2))⋊1C2, (C2×C40).63C22, C20.230(C4○D4), C4.114(C4○D20), C20.44D4⋊41C2, C20.48D4⋊41C2, C2.22(C8⋊D10), C10.73(C4⋊D4), C2.21(C20⋊7D4), C10.22(C8⋊C22), (C2×C20).775C23, (C2×D20).22C22, (C22×C10).103D4, (C22×C4).142D10, C22.134(C2×D20), C4⋊Dic5.26C22, C2.22(C8.D10), C10.22(C8.C22), (C22×C20).304C22, (C2×Dic10).21C22, (C2×C40⋊C2)⋊2C2, C4.113(C2×C5⋊D4), (C2×C10).165(C2×D4), (C2×C4).724(C22×D5), SmallGroup(320,761)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊2D4
G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a19, cbc=b-1 >
Subgroups: 550 in 120 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C40, C40, Dic10, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊D4, C40⋊C2, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C40, C5×M4(2), C2×Dic10, C2×D20, C2×C5⋊D4, C22×C20, C20.44D4, C40⋊5C4, D20⋊5C4, C2×C40⋊C2, C20.48D4, C20⋊7D4, C10×M4(2), C40⋊2D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, D20, C5⋊D4, C22×D5, C8⋊D4, C2×D20, C4○D20, C2×C5⋊D4, C8⋊D10, C8.D10, C20⋊7D4, C40⋊2D4
►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 56 130 100)(2 55 131 99)(3 54 132 98)(4 53 133 97)(5 52 134 96)(6 51 135 95)(7 50 136 94)(8 49 137 93)(9 48 138 92)(10 47 139 91)(11 46 140 90)(12 45 141 89)(13 44 142 88)(14 43 143 87)(15 42 144 86)(16 41 145 85)(17 80 146 84)(18 79 147 83)(19 78 148 82)(20 77 149 81)(21 76 150 120)(22 75 151 119)(23 74 152 118)(24 73 153 117)(25 72 154 116)(26 71 155 115)(27 70 156 114)(28 69 157 113)(29 68 158 112)(30 67 159 111)(31 66 160 110)(32 65 121 109)(33 64 122 108)(34 63 123 107)(35 62 124 106)(36 61 125 105)(37 60 126 104)(38 59 127 103)(39 58 128 102)(40 57 129 101)
(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 95)(42 114)(43 93)(44 112)(45 91)(46 110)(47 89)(48 108)(49 87)(50 106)(51 85)(52 104)(53 83)(54 102)(55 81)(56 100)(57 119)(58 98)(59 117)(60 96)(61 115)(62 94)(63 113)(64 92)(65 111)(66 90)(67 109)(68 88)(69 107)(70 86)(71 105)(72 84)(73 103)(74 82)(75 101)(76 120)(77 99)(78 118)(79 97)(80 116)(121 159)(122 138)(123 157)(124 136)(125 155)(126 134)(127 153)(128 132)(129 151)(131 149)(133 147)(135 145)(137 143)(139 141)(140 160)(142 158)(144 156)(146 154)(148 152)
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), (1,56,130,100)(2,55,131,99)(3,54,132,98)(4,53,133,97)(5,52,134,96)(6,51,135,95)(7,50,136,94)(8,49,137,93)(9,48,138,92)(10,47,139,91)(11,46,140,90)(12,45,141,89)(13,44,142,88)(14,43,143,87)(15,42,144,86)(16,41,145,85)(17,80,146,84)(18,79,147,83)(19,78,148,82)(20,77,149,81)(21,76,150,120)(22,75,151,119)(23,74,152,118)(24,73,153,117)(25,72,154,116)(26,71,155,115)(27,70,156,114)(28,69,157,113)(29,68,158,112)(30,67,159,111)(31,66,160,110)(32,65,121,109)(33,64,122,108)(34,63,123,107)(35,62,124,106)(36,61,125,105)(37,60,126,104)(38,59,127,103)(39,58,128,102)(40,57,129,101), (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)(41,95)(42,114)(43,93)(44,112)(45,91)(46,110)(47,89)(48,108)(49,87)(50,106)(51,85)(52,104)(53,83)(54,102)(55,81)(56,100)(57,119)(58,98)(59,117)(60,96)(61,115)(62,94)(63,113)(64,92)(65,111)(66,90)(67,109)(68,88)(69,107)(70,86)(71,105)(72,84)(73,103)(74,82)(75,101)(76,120)(77,99)(78,118)(79,97)(80,116)(121,159)(122,138)(123,157)(124,136)(125,155)(126,134)(127,153)(128,132)(129,151)(131,149)(133,147)(135,145)(137,143)(139,141)(140,160)(142,158)(144,156)(146,154)(148,152)>;
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), (1,56,130,100)(2,55,131,99)(3,54,132,98)(4,53,133,97)(5,52,134,96)(6,51,135,95)(7,50,136,94)(8,49,137,93)(9,48,138,92)(10,47,139,91)(11,46,140,90)(12,45,141,89)(13,44,142,88)(14,43,143,87)(15,42,144,86)(16,41,145,85)(17,80,146,84)(18,79,147,83)(19,78,148,82)(20,77,149,81)(21,76,150,120)(22,75,151,119)(23,74,152,118)(24,73,153,117)(25,72,154,116)(26,71,155,115)(27,70,156,114)(28,69,157,113)(29,68,158,112)(30,67,159,111)(31,66,160,110)(32,65,121,109)(33,64,122,108)(34,63,123,107)(35,62,124,106)(36,61,125,105)(37,60,126,104)(38,59,127,103)(39,58,128,102)(40,57,129,101), (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)(41,95)(42,114)(43,93)(44,112)(45,91)(46,110)(47,89)(48,108)(49,87)(50,106)(51,85)(52,104)(53,83)(54,102)(55,81)(56,100)(57,119)(58,98)(59,117)(60,96)(61,115)(62,94)(63,113)(64,92)(65,111)(66,90)(67,109)(68,88)(69,107)(70,86)(71,105)(72,84)(73,103)(74,82)(75,101)(76,120)(77,99)(78,118)(79,97)(80,116)(121,159)(122,138)(123,157)(124,136)(125,155)(126,134)(127,153)(128,132)(129,151)(131,149)(133,147)(135,145)(137,143)(139,141)(140,160)(142,158)(144,156)(146,154)(148,152) );
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)], [(1,56,130,100),(2,55,131,99),(3,54,132,98),(4,53,133,97),(5,52,134,96),(6,51,135,95),(7,50,136,94),(8,49,137,93),(9,48,138,92),(10,47,139,91),(11,46,140,90),(12,45,141,89),(13,44,142,88),(14,43,143,87),(15,42,144,86),(16,41,145,85),(17,80,146,84),(18,79,147,83),(19,78,148,82),(20,77,149,81),(21,76,150,120),(22,75,151,119),(23,74,152,118),(24,73,153,117),(25,72,154,116),(26,71,155,115),(27,70,156,114),(28,69,157,113),(29,68,158,112),(30,67,159,111),(31,66,160,110),(32,65,121,109),(33,64,122,108),(34,63,123,107),(35,62,124,106),(36,61,125,105),(37,60,126,104),(38,59,127,103),(39,58,128,102),(40,57,129,101)], [(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),(41,95),(42,114),(43,93),(44,112),(45,91),(46,110),(47,89),(48,108),(49,87),(50,106),(51,85),(52,104),(53,83),(54,102),(55,81),(56,100),(57,119),(58,98),(59,117),(60,96),(61,115),(62,94),(63,113),(64,92),(65,111),(66,90),(67,109),(68,88),(69,107),(70,86),(71,105),(72,84),(73,103),(74,82),(75,101),(76,120),(77,99),(78,118),(79,97),(80,116),(121,159),(122,138),(123,157),(124,136),(125,155),(126,134),(127,153),(128,132),(129,151),(131,149),(133,147),(135,145),(137,143),(139,141),(140,160),(142,158),(144,156),(146,154),(148,152)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 40 | 2 | 2 | 4 | 40 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | D20 | C4○D20 | C8⋊C22 | C8.C22 | C8⋊D10 | C8.D10 |
| kernel | C40⋊2D4 | C20.44D4 | C40⋊5C4 | D20⋊5C4 | C2×C40⋊C2 | C20.48D4 | C20⋊7D4 | C10×M4(2) | C40 | C2×C20 | C22×C10 | C2×M4(2) | C20 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C40⋊2D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 6 | 26 | 39 |
| 0 | 0 | 34 | 11 | 16 | 10 |
| 0 | 0 | 15 | 2 | 37 | 35 |
| 0 | 0 | 25 | 31 | 7 | 30 |
| 16 | 32 | 0 | 0 | 0 | 0 |
| 24 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 34 | 7 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,4,34,15,25,0,0,6,11,2,31,0,0,26,16,37,7,0,0,39,10,35,30],[16,24,0,0,0,0,32,25,0,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,34,34,0,0,0,0,1,7,0,0],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,1,7] >;
C40⋊2D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_2D_4 % in TeX
G:=Group("C40:2D4"); // GroupNames label
G:=SmallGroup(320,761);
// by ID
G=gap.SmallGroup(320,761);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations