metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊6D4, D10⋊3D8, (C2×D8)⋊5D5, (C10×D8)⋊6C2, C5⋊5(C8⋊7D4), C2.29(D5×D8), C20⋊2D4⋊4C2, C8⋊10(C5⋊D4), C40⋊5C4⋊23C2, C10.46(C2×D8), (C2×D4).64D10, C20.166(C2×D4), (C2×C8).238D10, C20.93(C4○D4), C10.34(C4○D8), D4⋊Dic5⋊29C2, (C2×C40).90C22, (C22×D5).89D4, C22.257(D4×D5), C4.28(D4⋊2D5), C2.18(D8⋊3D5), C2.16(C20⋊2D4), (C2×C20).434C23, (C2×Dic5).157D4, (D4×C10).83C22, C10.109(C4⋊D4), C4⋊Dic5.165C22, (D5×C2×C8)⋊3C2, C4.79(C2×C5⋊D4), (C2×C10).347(C2×D4), (C2×C4×D5).309C22, (C2×C4).524(C22×D5), (C2×C5⋊2C8).279C22, SmallGroup(320,784)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊6D4
G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >
Subgroups: 566 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C8⋊7D4, C8×D5, C2×C5⋊2C8, C4⋊Dic5, C23.D5, C2×C40, C5×D8, C2×C4×D5, C2×C5⋊D4, D4×C10, C40⋊5C4, D4⋊Dic5, D5×C2×C8, C20⋊2D4, C10×D8, C40⋊6D4
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C4○D8, C5⋊D4, C22×D5, C8⋊7D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D5×D8, D8⋊3D5, C20⋊2D4, C40⋊6D4
►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 42 138 109)(2 41 139 108)(3 80 140 107)(4 79 141 106)(5 78 142 105)(6 77 143 104)(7 76 144 103)(8 75 145 102)(9 74 146 101)(10 73 147 100)(11 72 148 99)(12 71 149 98)(13 70 150 97)(14 69 151 96)(15 68 152 95)(16 67 153 94)(17 66 154 93)(18 65 155 92)(19 64 156 91)(20 63 157 90)(21 62 158 89)(22 61 159 88)(23 60 160 87)(24 59 121 86)(25 58 122 85)(26 57 123 84)(27 56 124 83)(28 55 125 82)(29 54 126 81)(30 53 127 120)(31 52 128 119)(32 51 129 118)(33 50 130 117)(34 49 131 116)(35 48 132 115)(36 47 133 114)(37 46 134 113)(38 45 135 112)(39 44 136 111)(40 43 137 110)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(41 100)(42 109)(43 118)(44 87)(45 96)(46 105)(47 114)(48 83)(49 92)(50 101)(51 110)(52 119)(53 88)(54 97)(55 106)(56 115)(57 84)(58 93)(59 102)(60 111)(61 120)(62 89)(63 98)(64 107)(65 116)(66 85)(67 94)(68 103)(69 112)(70 81)(71 90)(72 99)(73 108)(74 117)(75 86)(76 95)(77 104)(78 113)(79 82)(80 91)(121 145)(122 154)(124 132)(125 141)(126 150)(127 159)(129 137)(130 146)(131 155)(134 142)(135 151)(136 160)(139 147)(140 156)(144 152)(149 157)
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), (1,42,138,109)(2,41,139,108)(3,80,140,107)(4,79,141,106)(5,78,142,105)(6,77,143,104)(7,76,144,103)(8,75,145,102)(9,74,146,101)(10,73,147,100)(11,72,148,99)(12,71,149,98)(13,70,150,97)(14,69,151,96)(15,68,152,95)(16,67,153,94)(17,66,154,93)(18,65,155,92)(19,64,156,91)(20,63,157,90)(21,62,158,89)(22,61,159,88)(23,60,160,87)(24,59,121,86)(25,58,122,85)(26,57,123,84)(27,56,124,83)(28,55,125,82)(29,54,126,81)(30,53,127,120)(31,52,128,119)(32,51,129,118)(33,50,130,117)(34,49,131,116)(35,48,132,115)(36,47,133,114)(37,46,134,113)(38,45,135,112)(39,44,136,111)(40,43,137,110), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(41,100)(42,109)(43,118)(44,87)(45,96)(46,105)(47,114)(48,83)(49,92)(50,101)(51,110)(52,119)(53,88)(54,97)(55,106)(56,115)(57,84)(58,93)(59,102)(60,111)(61,120)(62,89)(63,98)(64,107)(65,116)(66,85)(67,94)(68,103)(69,112)(70,81)(71,90)(72,99)(73,108)(74,117)(75,86)(76,95)(77,104)(78,113)(79,82)(80,91)(121,145)(122,154)(124,132)(125,141)(126,150)(127,159)(129,137)(130,146)(131,155)(134,142)(135,151)(136,160)(139,147)(140,156)(144,152)(149,157)>;
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), (1,42,138,109)(2,41,139,108)(3,80,140,107)(4,79,141,106)(5,78,142,105)(6,77,143,104)(7,76,144,103)(8,75,145,102)(9,74,146,101)(10,73,147,100)(11,72,148,99)(12,71,149,98)(13,70,150,97)(14,69,151,96)(15,68,152,95)(16,67,153,94)(17,66,154,93)(18,65,155,92)(19,64,156,91)(20,63,157,90)(21,62,158,89)(22,61,159,88)(23,60,160,87)(24,59,121,86)(25,58,122,85)(26,57,123,84)(27,56,124,83)(28,55,125,82)(29,54,126,81)(30,53,127,120)(31,52,128,119)(32,51,129,118)(33,50,130,117)(34,49,131,116)(35,48,132,115)(36,47,133,114)(37,46,134,113)(38,45,135,112)(39,44,136,111)(40,43,137,110), (2,10)(3,19)(4,28)(5,37)(7,15)(8,24)(9,33)(12,20)(13,29)(14,38)(17,25)(18,34)(22,30)(23,39)(27,35)(32,40)(41,100)(42,109)(43,118)(44,87)(45,96)(46,105)(47,114)(48,83)(49,92)(50,101)(51,110)(52,119)(53,88)(54,97)(55,106)(56,115)(57,84)(58,93)(59,102)(60,111)(61,120)(62,89)(63,98)(64,107)(65,116)(66,85)(67,94)(68,103)(69,112)(70,81)(71,90)(72,99)(73,108)(74,117)(75,86)(76,95)(77,104)(78,113)(79,82)(80,91)(121,145)(122,154)(124,132)(125,141)(126,150)(127,159)(129,137)(130,146)(131,155)(134,142)(135,151)(136,160)(139,147)(140,156)(144,152)(149,157) );
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)], [(1,42,138,109),(2,41,139,108),(3,80,140,107),(4,79,141,106),(5,78,142,105),(6,77,143,104),(7,76,144,103),(8,75,145,102),(9,74,146,101),(10,73,147,100),(11,72,148,99),(12,71,149,98),(13,70,150,97),(14,69,151,96),(15,68,152,95),(16,67,153,94),(17,66,154,93),(18,65,155,92),(19,64,156,91),(20,63,157,90),(21,62,158,89),(22,61,159,88),(23,60,160,87),(24,59,121,86),(25,58,122,85),(26,57,123,84),(27,56,124,83),(28,55,125,82),(29,54,126,81),(30,53,127,120),(31,52,128,119),(32,51,129,118),(33,50,130,117),(34,49,131,116),(35,48,132,115),(36,47,133,114),(37,46,134,113),(38,45,135,112),(39,44,136,111),(40,43,137,110)], [(2,10),(3,19),(4,28),(5,37),(7,15),(8,24),(9,33),(12,20),(13,29),(14,38),(17,25),(18,34),(22,30),(23,39),(27,35),(32,40),(41,100),(42,109),(43,118),(44,87),(45,96),(46,105),(47,114),(48,83),(49,92),(50,101),(51,110),(52,119),(53,88),(54,97),(55,106),(56,115),(57,84),(58,93),(59,102),(60,111),(61,120),(62,89),(63,98),(64,107),(65,116),(66,85),(67,94),(68,103),(69,112),(70,81),(71,90),(72,99),(73,108),(74,117),(75,86),(76,95),(77,104),(78,113),(79,82),(80,91),(121,145),(122,154),(124,132),(125,141),(126,150),(127,159),(129,137),(130,146),(131,155),(134,142),(135,151),(136,160),(139,147),(140,156),(144,152),(149,157)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 10 | 10 | 2 | 2 | 10 | 10 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D8 | D10 | D10 | C4○D8 | C5⋊D4 | D4⋊2D5 | D4×D5 | D5×D8 | D8⋊3D5 |
| kernel | C40⋊6D4 | C40⋊5C4 | D4⋊Dic5 | D5×C2×C8 | C20⋊2D4 | C10×D8 | C40 | C2×Dic5 | C22×D5 | C2×D8 | C20 | D10 | C2×C8 | C2×D4 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊6D4 ►in GL4(𝔽41) generated by
| 35 | 7 | 0 | 0 |
| 35 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 24 | 14 |
| 21 | 17 | 0 | 0 |
| 15 | 20 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 40 |
| 6 | 1 | 0 | 0 |
| 6 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [35,35,0,0,7,0,0,0,0,0,3,24,0,0,0,14],[21,15,0,0,17,20,0,0,0,0,1,0,0,0,9,40],[6,6,0,0,1,35,0,0,0,0,1,0,0,0,0,1] >;
C40⋊6D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_6D_4 % in TeX
G:=Group("C40:6D4"); // GroupNames label
G:=SmallGroup(320,784);
// by ID
G=gap.SmallGroup(320,784);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations