metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.43D4, C4.24(D4×D5), (C2×SD16)⋊9D5, C5⋊2C8.32D4, (C10×SD16)⋊7C2, (C8×Dic5)⋊10C2, (C2×D4).71D10, C20.175(C2×D4), (C2×C8).262D10, C5⋊4(C8.12D4), C8.20(C5⋊D4), (C2×Q8).53D10, C10.62(C4○D8), C20.23D4⋊4C2, C20.17D4⋊6C2, C22.265(D4×D5), C10.30(C4⋊1D4), C2.21(C20⋊D4), (C2×C20).445C23, (C2×C40).163C22, (C2×Dic5).158D4, (D4×C10).94C22, (Q8×C10).75C22, (C2×D20).123C22, C2.28(SD16⋊3D5), (C4×Dic5).272C22, (C2×Dic10).130C22, C4.8(C2×C5⋊D4), (C2×D4⋊D5).9C2, (C2×C40⋊C2)⋊29C2, (C2×C5⋊Q16)⋊18C2, (C2×C10).357(C2×D4), (C2×C4).534(C22×D5), (C2×C5⋊2C8).281C22, SmallGroup(320,795)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.43D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=a9, cac-1=a19, cbc-1=a20b-1 >
Subgroups: 558 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×SD16, C2×Q16, C5⋊2C8, C40, Dic10, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C8.12D4, C40⋊C2, C2×C5⋊2C8, C4×Dic5, D10⋊C4, D4⋊D5, C5⋊Q16, C23.D5, C2×C40, C5×SD16, C2×Dic10, C2×D20, D4×C10, Q8×C10, C8×Dic5, C2×C40⋊C2, C2×D4⋊D5, C20.17D4, C2×C5⋊Q16, C20.23D4, C10×SD16, C40.43D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C4○D8, C5⋊D4, C22×D5, C8.12D4, D4×D5, C2×C5⋊D4, SD16⋊3D5, C20⋊D4, C40.43D4
►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)
(1 148 43 95)(2 157 44 104)(3 126 45 113)(4 135 46 82)(5 144 47 91)(6 153 48 100)(7 122 49 109)(8 131 50 118)(9 140 51 87)(10 149 52 96)(11 158 53 105)(12 127 54 114)(13 136 55 83)(14 145 56 92)(15 154 57 101)(16 123 58 110)(17 132 59 119)(18 141 60 88)(19 150 61 97)(20 159 62 106)(21 128 63 115)(22 137 64 84)(23 146 65 93)(24 155 66 102)(25 124 67 111)(26 133 68 120)(27 142 69 89)(28 151 70 98)(29 160 71 107)(30 129 72 116)(31 138 73 85)(32 147 74 94)(33 156 75 103)(34 125 76 112)(35 134 77 81)(36 143 78 90)(37 152 79 99)(38 121 80 108)(39 130 41 117)(40 139 42 86)
(1 16 21 36)(2 35 22 15)(3 14 23 34)(4 33 24 13)(5 12 25 32)(6 31 26 11)(7 10 27 30)(8 29 28 9)(17 40 37 20)(18 19 38 39)(41 60 61 80)(42 79 62 59)(43 58 63 78)(44 77 64 57)(45 56 65 76)(46 75 66 55)(47 54 67 74)(48 73 68 53)(49 52 69 72)(50 71 70 51)(81 157 101 137)(82 136 102 156)(83 155 103 135)(84 134 104 154)(85 153 105 133)(86 132 106 152)(87 151 107 131)(88 130 108 150)(89 149 109 129)(90 128 110 148)(91 147 111 127)(92 126 112 146)(93 145 113 125)(94 124 114 144)(95 143 115 123)(96 122 116 142)(97 141 117 121)(98 160 118 140)(99 139 119 159)(100 158 120 138)
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), (1,148,43,95)(2,157,44,104)(3,126,45,113)(4,135,46,82)(5,144,47,91)(6,153,48,100)(7,122,49,109)(8,131,50,118)(9,140,51,87)(10,149,52,96)(11,158,53,105)(12,127,54,114)(13,136,55,83)(14,145,56,92)(15,154,57,101)(16,123,58,110)(17,132,59,119)(18,141,60,88)(19,150,61,97)(20,159,62,106)(21,128,63,115)(22,137,64,84)(23,146,65,93)(24,155,66,102)(25,124,67,111)(26,133,68,120)(27,142,69,89)(28,151,70,98)(29,160,71,107)(30,129,72,116)(31,138,73,85)(32,147,74,94)(33,156,75,103)(34,125,76,112)(35,134,77,81)(36,143,78,90)(37,152,79,99)(38,121,80,108)(39,130,41,117)(40,139,42,86), (1,16,21,36)(2,35,22,15)(3,14,23,34)(4,33,24,13)(5,12,25,32)(6,31,26,11)(7,10,27,30)(8,29,28,9)(17,40,37,20)(18,19,38,39)(41,60,61,80)(42,79,62,59)(43,58,63,78)(44,77,64,57)(45,56,65,76)(46,75,66,55)(47,54,67,74)(48,73,68,53)(49,52,69,72)(50,71,70,51)(81,157,101,137)(82,136,102,156)(83,155,103,135)(84,134,104,154)(85,153,105,133)(86,132,106,152)(87,151,107,131)(88,130,108,150)(89,149,109,129)(90,128,110,148)(91,147,111,127)(92,126,112,146)(93,145,113,125)(94,124,114,144)(95,143,115,123)(96,122,116,142)(97,141,117,121)(98,160,118,140)(99,139,119,159)(100,158,120,138)>;
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), (1,148,43,95)(2,157,44,104)(3,126,45,113)(4,135,46,82)(5,144,47,91)(6,153,48,100)(7,122,49,109)(8,131,50,118)(9,140,51,87)(10,149,52,96)(11,158,53,105)(12,127,54,114)(13,136,55,83)(14,145,56,92)(15,154,57,101)(16,123,58,110)(17,132,59,119)(18,141,60,88)(19,150,61,97)(20,159,62,106)(21,128,63,115)(22,137,64,84)(23,146,65,93)(24,155,66,102)(25,124,67,111)(26,133,68,120)(27,142,69,89)(28,151,70,98)(29,160,71,107)(30,129,72,116)(31,138,73,85)(32,147,74,94)(33,156,75,103)(34,125,76,112)(35,134,77,81)(36,143,78,90)(37,152,79,99)(38,121,80,108)(39,130,41,117)(40,139,42,86), (1,16,21,36)(2,35,22,15)(3,14,23,34)(4,33,24,13)(5,12,25,32)(6,31,26,11)(7,10,27,30)(8,29,28,9)(17,40,37,20)(18,19,38,39)(41,60,61,80)(42,79,62,59)(43,58,63,78)(44,77,64,57)(45,56,65,76)(46,75,66,55)(47,54,67,74)(48,73,68,53)(49,52,69,72)(50,71,70,51)(81,157,101,137)(82,136,102,156)(83,155,103,135)(84,134,104,154)(85,153,105,133)(86,132,106,152)(87,151,107,131)(88,130,108,150)(89,149,109,129)(90,128,110,148)(91,147,111,127)(92,126,112,146)(93,145,113,125)(94,124,114,144)(95,143,115,123)(96,122,116,142)(97,141,117,121)(98,160,118,140)(99,139,119,159)(100,158,120,138) );
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)], [(1,148,43,95),(2,157,44,104),(3,126,45,113),(4,135,46,82),(5,144,47,91),(6,153,48,100),(7,122,49,109),(8,131,50,118),(9,140,51,87),(10,149,52,96),(11,158,53,105),(12,127,54,114),(13,136,55,83),(14,145,56,92),(15,154,57,101),(16,123,58,110),(17,132,59,119),(18,141,60,88),(19,150,61,97),(20,159,62,106),(21,128,63,115),(22,137,64,84),(23,146,65,93),(24,155,66,102),(25,124,67,111),(26,133,68,120),(27,142,69,89),(28,151,70,98),(29,160,71,107),(30,129,72,116),(31,138,73,85),(32,147,74,94),(33,156,75,103),(34,125,76,112),(35,134,77,81),(36,143,78,90),(37,152,79,99),(38,121,80,108),(39,130,41,117),(40,139,42,86)], [(1,16,21,36),(2,35,22,15),(3,14,23,34),(4,33,24,13),(5,12,25,32),(6,31,26,11),(7,10,27,30),(8,29,28,9),(17,40,37,20),(18,19,38,39),(41,60,61,80),(42,79,62,59),(43,58,63,78),(44,77,64,57),(45,56,65,76),(46,75,66,55),(47,54,67,74),(48,73,68,53),(49,52,69,72),(50,71,70,51),(81,157,101,137),(82,136,102,156),(83,155,103,135),(84,134,104,154),(85,153,105,133),(86,132,106,152),(87,151,107,131),(88,130,108,150),(89,149,109,129),(90,128,110,148),(91,147,111,127),(92,126,112,146),(93,145,113,125),(94,124,114,144),(95,143,115,123),(96,122,116,142),(97,141,117,121),(98,160,118,140),(99,139,119,159),(100,158,120,138)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 40 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | D4×D5 | D4×D5 | SD16⋊3D5 |
| kernel | C40.43D4 | C8×Dic5 | C2×C40⋊C2 | C2×D4⋊D5 | C20.17D4 | C2×C5⋊Q16 | C20.23D4 | C10×SD16 | C5⋊2C8 | C40 | C2×Dic5 | C2×SD16 | C2×C8 | C2×D4 | C2×Q8 | C10 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 8 |
Matrix representation of C40.43D4 ►in GL6(𝔽41)
| 25 | 36 | 0 | 0 | 0 | 0 |
| 35 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 7 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 25 | 36 | 0 | 0 | 0 | 0 |
| 35 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 25 | 36 | 0 | 0 | 0 | 0 |
| 10 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 0 | 15 | 30 |
G:=sub<GL(6,GF(41))| [25,35,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,7,0,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[25,35,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[25,10,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,11,15,0,0,0,0,11,30] >;
C40.43D4 in GAP, Magma, Sage, TeX
C_{40}._{43}D_4 % in TeX
G:=Group("C40.43D4"); // GroupNames label
G:=SmallGroup(320,795);
// by ID
G=gap.SmallGroup(320,795);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^19,c*b*c^-1=a^20*b^-1>;
// generators/relations