metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C10)⋊3Q16, C4⋊C4.70D10, C4.176(D4×D5), C5⋊2C8.45D4, C22⋊Q8.6D5, (C2×C20).268D4, C20.157(C2×D4), C5⋊3(C8.18D4), (C2×Q8).32D10, C10.39(C2×Q16), C10.Q16⋊39C2, C10.D8⋊41C2, Q8⋊Dic5⋊17C2, C22⋊1(C5⋊Q16), (C22×C10).97D4, C10.102(C4○D8), C20.192(C4○D4), C4.65(D4⋊2D5), C10.99(C4⋊D4), (C2×C20).370C23, (C22×C4).343D10, C23.42(C5⋊D4), (Q8×C10).50C22, C20.48D4.12C2, C4⋊Dic5.148C22, C2.20(Dic5⋊D4), C2.21(D4.8D10), (C22×C20).174C22, (C2×Dic10).109C22, (C2×C5⋊Q16)⋊10C2, C2.10(C2×C5⋊Q16), (C5×C22⋊Q8).5C2, (C2×C10).501(C2×D4), (C22×C5⋊2C8).9C2, (C2×C4).108(C5⋊D4), (C5×C4⋊C4).117C22, (C2×C4).470(C22×D5), C22.176(C2×C5⋊D4), (C2×C5⋊2C8).259C22, SmallGroup(320,678)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C10)⋊Q16
G = < a,b,c,d | a2=b10=c8=1, d2=c4, ab=ba, ac=ca, dad-1=ab5, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 358 in 114 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, Q8⋊C4, C2.D8, C22⋊Q8, C22⋊Q8, C22×C8, C2×Q16, C5⋊2C8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×C10, C8.18D4, C2×C5⋊2C8, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C5⋊Q16, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, Q8×C10, C10.D8, C10.Q16, Q8⋊Dic5, C22×C5⋊2C8, C20.48D4, C2×C5⋊Q16, C5×C22⋊Q8, (C2×C10)⋊Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C4○D8, C5⋊D4, C22×D5, C8.18D4, C5⋊Q16, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊D4, C2×C5⋊Q16, D4.8D10, (C2×C10)⋊Q16
►On 160 points
Generators in S160
(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)(121 126)(122 127)(123 128)(124 129)(125 130)
(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)
(1 160 36 145 30 20 50 140)(2 159 37 144 21 19 41 139)(3 158 38 143 22 18 42 138)(4 157 39 142 23 17 43 137)(5 156 40 141 24 16 44 136)(6 155 31 150 25 15 45 135)(7 154 32 149 26 14 46 134)(8 153 33 148 27 13 47 133)(9 152 34 147 28 12 48 132)(10 151 35 146 29 11 49 131)(51 115 86 95 65 130 71 110)(52 114 87 94 66 129 72 109)(53 113 88 93 67 128 73 108)(54 112 89 92 68 127 74 107)(55 111 90 91 69 126 75 106)(56 120 81 100 70 125 76 105)(57 119 82 99 61 124 77 104)(58 118 83 98 62 123 78 103)(59 117 84 97 63 122 79 102)(60 116 85 96 64 121 80 101)
(1 70 30 56)(2 61 21 57)(3 62 22 58)(4 63 23 59)(5 64 24 60)(6 65 25 51)(7 66 26 52)(8 67 27 53)(9 68 28 54)(10 69 29 55)(11 106 151 91)(12 107 152 92)(13 108 153 93)(14 109 154 94)(15 110 155 95)(16 101 156 96)(17 102 157 97)(18 103 158 98)(19 104 159 99)(20 105 160 100)(31 86 45 71)(32 87 46 72)(33 88 47 73)(34 89 48 74)(35 90 49 75)(36 81 50 76)(37 82 41 77)(38 83 42 78)(39 84 43 79)(40 85 44 80)(111 131 126 146)(112 132 127 147)(113 133 128 148)(114 134 129 149)(115 135 130 150)(116 136 121 141)(117 137 122 142)(118 138 123 143)(119 139 124 144)(120 140 125 145)
G:=sub<Sym(160)| (51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130), (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), (1,160,36,145,30,20,50,140)(2,159,37,144,21,19,41,139)(3,158,38,143,22,18,42,138)(4,157,39,142,23,17,43,137)(5,156,40,141,24,16,44,136)(6,155,31,150,25,15,45,135)(7,154,32,149,26,14,46,134)(8,153,33,148,27,13,47,133)(9,152,34,147,28,12,48,132)(10,151,35,146,29,11,49,131)(51,115,86,95,65,130,71,110)(52,114,87,94,66,129,72,109)(53,113,88,93,67,128,73,108)(54,112,89,92,68,127,74,107)(55,111,90,91,69,126,75,106)(56,120,81,100,70,125,76,105)(57,119,82,99,61,124,77,104)(58,118,83,98,62,123,78,103)(59,117,84,97,63,122,79,102)(60,116,85,96,64,121,80,101), (1,70,30,56)(2,61,21,57)(3,62,22,58)(4,63,23,59)(5,64,24,60)(6,65,25,51)(7,66,26,52)(8,67,27,53)(9,68,28,54)(10,69,29,55)(11,106,151,91)(12,107,152,92)(13,108,153,93)(14,109,154,94)(15,110,155,95)(16,101,156,96)(17,102,157,97)(18,103,158,98)(19,104,159,99)(20,105,160,100)(31,86,45,71)(32,87,46,72)(33,88,47,73)(34,89,48,74)(35,90,49,75)(36,81,50,76)(37,82,41,77)(38,83,42,78)(39,84,43,79)(40,85,44,80)(111,131,126,146)(112,132,127,147)(113,133,128,148)(114,134,129,149)(115,135,130,150)(116,136,121,141)(117,137,122,142)(118,138,123,143)(119,139,124,144)(120,140,125,145)>;
G:=Group( (51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120)(121,126)(122,127)(123,128)(124,129)(125,130), (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), (1,160,36,145,30,20,50,140)(2,159,37,144,21,19,41,139)(3,158,38,143,22,18,42,138)(4,157,39,142,23,17,43,137)(5,156,40,141,24,16,44,136)(6,155,31,150,25,15,45,135)(7,154,32,149,26,14,46,134)(8,153,33,148,27,13,47,133)(9,152,34,147,28,12,48,132)(10,151,35,146,29,11,49,131)(51,115,86,95,65,130,71,110)(52,114,87,94,66,129,72,109)(53,113,88,93,67,128,73,108)(54,112,89,92,68,127,74,107)(55,111,90,91,69,126,75,106)(56,120,81,100,70,125,76,105)(57,119,82,99,61,124,77,104)(58,118,83,98,62,123,78,103)(59,117,84,97,63,122,79,102)(60,116,85,96,64,121,80,101), (1,70,30,56)(2,61,21,57)(3,62,22,58)(4,63,23,59)(5,64,24,60)(6,65,25,51)(7,66,26,52)(8,67,27,53)(9,68,28,54)(10,69,29,55)(11,106,151,91)(12,107,152,92)(13,108,153,93)(14,109,154,94)(15,110,155,95)(16,101,156,96)(17,102,157,97)(18,103,158,98)(19,104,159,99)(20,105,160,100)(31,86,45,71)(32,87,46,72)(33,88,47,73)(34,89,48,74)(35,90,49,75)(36,81,50,76)(37,82,41,77)(38,83,42,78)(39,84,43,79)(40,85,44,80)(111,131,126,146)(112,132,127,147)(113,133,128,148)(114,134,129,149)(115,135,130,150)(116,136,121,141)(117,137,122,142)(118,138,123,143)(119,139,124,144)(120,140,125,145) );
G=PermutationGroup([[(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80),(81,86),(82,87),(83,88),(84,89),(85,90),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110),(111,116),(112,117),(113,118),(114,119),(115,120),(121,126),(122,127),(123,128),(124,129),(125,130)], [(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)], [(1,160,36,145,30,20,50,140),(2,159,37,144,21,19,41,139),(3,158,38,143,22,18,42,138),(4,157,39,142,23,17,43,137),(5,156,40,141,24,16,44,136),(6,155,31,150,25,15,45,135),(7,154,32,149,26,14,46,134),(8,153,33,148,27,13,47,133),(9,152,34,147,28,12,48,132),(10,151,35,146,29,11,49,131),(51,115,86,95,65,130,71,110),(52,114,87,94,66,129,72,109),(53,113,88,93,67,128,73,108),(54,112,89,92,68,127,74,107),(55,111,90,91,69,126,75,106),(56,120,81,100,70,125,76,105),(57,119,82,99,61,124,77,104),(58,118,83,98,62,123,78,103),(59,117,84,97,63,122,79,102),(60,116,85,96,64,121,80,101)], [(1,70,30,56),(2,61,21,57),(3,62,22,58),(4,63,23,59),(5,64,24,60),(6,65,25,51),(7,66,26,52),(8,67,27,53),(9,68,28,54),(10,69,29,55),(11,106,151,91),(12,107,152,92),(13,108,153,93),(14,109,154,94),(15,110,155,95),(16,101,156,96),(17,102,157,97),(18,103,158,98),(19,104,159,99),(20,105,160,100),(31,86,45,71),(32,87,46,72),(33,88,47,73),(34,89,48,74),(35,90,49,75),(36,81,50,76),(37,82,41,77),(38,83,42,78),(39,84,43,79),(40,85,44,80),(111,131,126,146),(112,132,127,147),(113,133,128,148),(114,134,129,149),(115,135,130,150),(116,136,121,141),(117,137,122,142),(118,138,123,143),(119,139,124,144),(120,140,125,145)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | Q16 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | D4×D5 | D4⋊2D5 | C5⋊Q16 | D4.8D10 |
| kernel | (C2×C10)⋊Q16 | C10.D8 | C10.Q16 | Q8⋊Dic5 | C22×C5⋊2C8 | C20.48D4 | C2×C5⋊Q16 | C5×C22⋊Q8 | C5⋊2C8 | C2×C20 | C22×C10 | C22⋊Q8 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C2×C4 | C23 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of (C2×C10)⋊Q16 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 22 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 6 |
| 0 | 0 | 0 | 0 | 34 | 24 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34],[38,0,0,0,0,0,0,27,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,22,0,0,0,0,13,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,34,0,0,0,0,6,24] >;
(C2×C10)⋊Q16 in GAP, Magma, Sage, TeX
(C_2\times C_{10})\rtimes Q_{16} % in TeX
G:=Group("(C2xC10):Q16"); // GroupNames label
G:=SmallGroup(320,678);
// by ID
G=gap.SmallGroup(320,678);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,219,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^5,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations