metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊Q8⋊4C2, C4⋊C4.6D10, C40⋊8C4⋊17C2, D4⋊C4.8D5, (C2×D4).20D10, C20.4(C4○D4), (C2×C8).167D10, C10.Q16⋊1C2, C4.22(C4○D20), (C2×Dic5).24D4, D4⋊Dic5.4C2, C22.167(D4×D5), C2.12(D8⋊D5), C4.48(D4⋊2D5), C20.44D4⋊16C2, C10.28(C8⋊C22), (C2×C20).205C23, (C2×C40).180C22, C20.17D4.4C2, C2.9(SD16⋊D5), (D4×C10).26C22, C4⋊Dic5.64C22, C10.24(C4.4D4), C10.26(C8.C22), (C4×Dic5).15C22, C5⋊2(C42.28C22), (C2×Dic10).56C22, C2.14(Dic5.5D4), (C5×D4⋊C4).9C2, (C2×C10).218(C2×D4), (C5×C4⋊C4).10C22, (C2×C5⋊2C8).11C22, (C2×C4).312(C22×D5), SmallGroup(320,392)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊Q8⋊C2
G = < a,b,c,d | a20=b4=d2=1, c2=b2, bab-1=dad=a11, cac-1=a9, cbc-1=b-1, dbd=a15b-1, dcd=a10b2c >
Subgroups: 398 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C42.28C22, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, D4×C10, C10.Q16, C40⋊8C4, C20.44D4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, C20.17D4, C20⋊Q8⋊C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, C8.C22, C22×D5, C42.28C22, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, D8⋊D5, SD16⋊D5, C20⋊Q8⋊C2
►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 54 101 92)(2 45 102 83)(3 56 103 94)(4 47 104 85)(5 58 105 96)(6 49 106 87)(7 60 107 98)(8 51 108 89)(9 42 109 100)(10 53 110 91)(11 44 111 82)(12 55 112 93)(13 46 113 84)(14 57 114 95)(15 48 115 86)(16 59 116 97)(17 50 117 88)(18 41 118 99)(19 52 119 90)(20 43 120 81)(21 141 67 137)(22 152 68 128)(23 143 69 139)(24 154 70 130)(25 145 71 121)(26 156 72 132)(27 147 73 123)(28 158 74 134)(29 149 75 125)(30 160 76 136)(31 151 77 127)(32 142 78 138)(33 153 79 129)(34 144 80 140)(35 155 61 131)(36 146 62 122)(37 157 63 133)(38 148 64 124)(39 159 65 135)(40 150 66 126)
(1 155 101 131)(2 144 102 140)(3 153 103 129)(4 142 104 138)(5 151 105 127)(6 160 106 136)(7 149 107 125)(8 158 108 134)(9 147 109 123)(10 156 110 132)(11 145 111 121)(12 154 112 130)(13 143 113 139)(14 152 114 128)(15 141 115 137)(16 150 116 126)(17 159 117 135)(18 148 118 124)(19 157 119 133)(20 146 120 122)(21 86 67 48)(22 95 68 57)(23 84 69 46)(24 93 70 55)(25 82 71 44)(26 91 72 53)(27 100 73 42)(28 89 74 51)(29 98 75 60)(30 87 76 49)(31 96 77 58)(32 85 78 47)(33 94 79 56)(34 83 80 45)(35 92 61 54)(36 81 62 43)(37 90 63 52)(38 99 64 41)(39 88 65 50)(40 97 66 59)
(2 12)(4 14)(6 16)(8 18)(10 20)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)(41 94)(42 85)(43 96)(44 87)(45 98)(46 89)(47 100)(48 91)(49 82)(50 93)(51 84)(52 95)(53 86)(54 97)(55 88)(56 99)(57 90)(58 81)(59 92)(60 83)(61 76)(62 67)(63 78)(64 69)(65 80)(66 71)(68 73)(70 75)(72 77)(74 79)(102 112)(104 114)(106 116)(108 118)(110 120)(121 155)(122 146)(123 157)(124 148)(125 159)(126 150)(127 141)(128 152)(129 143)(130 154)(131 145)(132 156)(133 147)(134 158)(135 149)(136 160)(137 151)(138 142)(139 153)(140 144)
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), (1,54,101,92)(2,45,102,83)(3,56,103,94)(4,47,104,85)(5,58,105,96)(6,49,106,87)(7,60,107,98)(8,51,108,89)(9,42,109,100)(10,53,110,91)(11,44,111,82)(12,55,112,93)(13,46,113,84)(14,57,114,95)(15,48,115,86)(16,59,116,97)(17,50,117,88)(18,41,118,99)(19,52,119,90)(20,43,120,81)(21,141,67,137)(22,152,68,128)(23,143,69,139)(24,154,70,130)(25,145,71,121)(26,156,72,132)(27,147,73,123)(28,158,74,134)(29,149,75,125)(30,160,76,136)(31,151,77,127)(32,142,78,138)(33,153,79,129)(34,144,80,140)(35,155,61,131)(36,146,62,122)(37,157,63,133)(38,148,64,124)(39,159,65,135)(40,150,66,126), (1,155,101,131)(2,144,102,140)(3,153,103,129)(4,142,104,138)(5,151,105,127)(6,160,106,136)(7,149,107,125)(8,158,108,134)(9,147,109,123)(10,156,110,132)(11,145,111,121)(12,154,112,130)(13,143,113,139)(14,152,114,128)(15,141,115,137)(16,150,116,126)(17,159,117,135)(18,148,118,124)(19,157,119,133)(20,146,120,122)(21,86,67,48)(22,95,68,57)(23,84,69,46)(24,93,70,55)(25,82,71,44)(26,91,72,53)(27,100,73,42)(28,89,74,51)(29,98,75,60)(30,87,76,49)(31,96,77,58)(32,85,78,47)(33,94,79,56)(34,83,80,45)(35,92,61,54)(36,81,62,43)(37,90,63,52)(38,99,64,41)(39,88,65,50)(40,97,66,59), (2,12)(4,14)(6,16)(8,18)(10,20)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,94)(42,85)(43,96)(44,87)(45,98)(46,89)(47,100)(48,91)(49,82)(50,93)(51,84)(52,95)(53,86)(54,97)(55,88)(56,99)(57,90)(58,81)(59,92)(60,83)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)(102,112)(104,114)(106,116)(108,118)(110,120)(121,155)(122,146)(123,157)(124,148)(125,159)(126,150)(127,141)(128,152)(129,143)(130,154)(131,145)(132,156)(133,147)(134,158)(135,149)(136,160)(137,151)(138,142)(139,153)(140,144)>;
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), (1,54,101,92)(2,45,102,83)(3,56,103,94)(4,47,104,85)(5,58,105,96)(6,49,106,87)(7,60,107,98)(8,51,108,89)(9,42,109,100)(10,53,110,91)(11,44,111,82)(12,55,112,93)(13,46,113,84)(14,57,114,95)(15,48,115,86)(16,59,116,97)(17,50,117,88)(18,41,118,99)(19,52,119,90)(20,43,120,81)(21,141,67,137)(22,152,68,128)(23,143,69,139)(24,154,70,130)(25,145,71,121)(26,156,72,132)(27,147,73,123)(28,158,74,134)(29,149,75,125)(30,160,76,136)(31,151,77,127)(32,142,78,138)(33,153,79,129)(34,144,80,140)(35,155,61,131)(36,146,62,122)(37,157,63,133)(38,148,64,124)(39,159,65,135)(40,150,66,126), (1,155,101,131)(2,144,102,140)(3,153,103,129)(4,142,104,138)(5,151,105,127)(6,160,106,136)(7,149,107,125)(8,158,108,134)(9,147,109,123)(10,156,110,132)(11,145,111,121)(12,154,112,130)(13,143,113,139)(14,152,114,128)(15,141,115,137)(16,150,116,126)(17,159,117,135)(18,148,118,124)(19,157,119,133)(20,146,120,122)(21,86,67,48)(22,95,68,57)(23,84,69,46)(24,93,70,55)(25,82,71,44)(26,91,72,53)(27,100,73,42)(28,89,74,51)(29,98,75,60)(30,87,76,49)(31,96,77,58)(32,85,78,47)(33,94,79,56)(34,83,80,45)(35,92,61,54)(36,81,62,43)(37,90,63,52)(38,99,64,41)(39,88,65,50)(40,97,66,59), (2,12)(4,14)(6,16)(8,18)(10,20)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,94)(42,85)(43,96)(44,87)(45,98)(46,89)(47,100)(48,91)(49,82)(50,93)(51,84)(52,95)(53,86)(54,97)(55,88)(56,99)(57,90)(58,81)(59,92)(60,83)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)(102,112)(104,114)(106,116)(108,118)(110,120)(121,155)(122,146)(123,157)(124,148)(125,159)(126,150)(127,141)(128,152)(129,143)(130,154)(131,145)(132,156)(133,147)(134,158)(135,149)(136,160)(137,151)(138,142)(139,153)(140,144) );
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)], [(1,54,101,92),(2,45,102,83),(3,56,103,94),(4,47,104,85),(5,58,105,96),(6,49,106,87),(7,60,107,98),(8,51,108,89),(9,42,109,100),(10,53,110,91),(11,44,111,82),(12,55,112,93),(13,46,113,84),(14,57,114,95),(15,48,115,86),(16,59,116,97),(17,50,117,88),(18,41,118,99),(19,52,119,90),(20,43,120,81),(21,141,67,137),(22,152,68,128),(23,143,69,139),(24,154,70,130),(25,145,71,121),(26,156,72,132),(27,147,73,123),(28,158,74,134),(29,149,75,125),(30,160,76,136),(31,151,77,127),(32,142,78,138),(33,153,79,129),(34,144,80,140),(35,155,61,131),(36,146,62,122),(37,157,63,133),(38,148,64,124),(39,159,65,135),(40,150,66,126)], [(1,155,101,131),(2,144,102,140),(3,153,103,129),(4,142,104,138),(5,151,105,127),(6,160,106,136),(7,149,107,125),(8,158,108,134),(9,147,109,123),(10,156,110,132),(11,145,111,121),(12,154,112,130),(13,143,113,139),(14,152,114,128),(15,141,115,137),(16,150,116,126),(17,159,117,135),(18,148,118,124),(19,157,119,133),(20,146,120,122),(21,86,67,48),(22,95,68,57),(23,84,69,46),(24,93,70,55),(25,82,71,44),(26,91,72,53),(27,100,73,42),(28,89,74,51),(29,98,75,60),(30,87,76,49),(31,96,77,58),(32,85,78,47),(33,94,79,56),(34,83,80,45),(35,92,61,54),(36,81,62,43),(37,90,63,52),(38,99,64,41),(39,88,65,50),(40,97,66,59)], [(2,12),(4,14),(6,16),(8,18),(10,20),(21,36),(22,27),(23,38),(24,29),(25,40),(26,31),(28,33),(30,35),(32,37),(34,39),(41,94),(42,85),(43,96),(44,87),(45,98),(46,89),(47,100),(48,91),(49,82),(50,93),(51,84),(52,95),(53,86),(54,97),(55,88),(56,99),(57,90),(58,81),(59,92),(60,83),(61,76),(62,67),(63,78),(64,69),(65,80),(66,71),(68,73),(70,75),(72,77),(74,79),(102,112),(104,114),(106,116),(108,118),(110,120),(121,155),(122,146),(123,157),(124,148),(125,159),(126,150),(127,141),(128,152),(129,143),(130,154),(131,145),(132,156),(133,147),(134,158),(135,149),(136,160),(137,151),(138,142),(139,153),(140,144)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 2 | 2 | 8 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8⋊C22 | C8.C22 | D4⋊2D5 | D4×D5 | D8⋊D5 | SD16⋊D5 |
| kernel | C20⋊Q8⋊C2 | C10.Q16 | C40⋊8C4 | C20.44D4 | D4⋊Dic5 | C5×D4⋊C4 | C20⋊Q8 | C20.17D4 | C2×Dic5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C20⋊Q8⋊C2 ►in GL8(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 | 28 | 37 |
| 0 | 0 | 0 | 0 | 33 | 40 | 13 | 32 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 21 | 3 | 1 | 35 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 21 | 7 | 20 |
| 0 | 0 | 0 | 0 | 37 | 39 | 13 | 28 |
| 0 | 0 | 0 | 0 | 3 | 33 | 22 | 20 |
| 0 | 0 | 0 | 0 | 3 | 11 | 21 | 19 |
| 0 | 0 | 38 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 3 | 0 | 0 | 0 | 0 |
| 38 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 38 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 | 14 | 27 |
| 0 | 0 | 0 | 0 | 37 | 40 | 13 | 2 |
| 0 | 0 | 0 | 0 | 9 | 22 | 21 | 18 |
| 0 | 0 | 0 | 0 | 3 | 22 | 21 | 20 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 38 | 40 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,7,33,3,21,0,0,0,0,1,40,0,3,0,0,0,0,28,13,0,1,0,0,0,0,37,32,40,35],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,37,3,3,0,0,0,0,21,39,33,11,0,0,0,0,7,13,22,21,0,0,0,0,20,28,20,19],[0,0,38,38,0,0,0,0,0,0,17,3,0,0,0,0,38,38,0,0,0,0,0,0,17,3,0,0,0,0,0,0,0,0,0,0,1,37,9,3,0,0,0,0,21,40,22,22,0,0,0,0,14,13,21,21,0,0,0,0,27,2,18,20],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,38,3,0,0,0,0,0,1,38,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
C20⋊Q8⋊C2 in GAP, Magma, Sage, TeX
C_{20}\rtimes Q_8\rtimes C_2 % in TeX
G:=Group("C20:Q8:C2"); // GroupNames label
G:=SmallGroup(320,392);
// by ID
G=gap.SmallGroup(320,392);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,1094,135,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^4=d^2=1,c^2=b^2,b*a*b^-1=d*a*d=a^11,c*a*c^-1=a^9,c*b*c^-1=b^-1,d*b*d=a^15*b^-1,d*c*d=a^10*b^2*c>;
// generators/relations