direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊D20, C42⋊39D10, (C2×C4)⋊7D20, C4⋊2(C2×D20), (C2×C20)⋊30D4, C20⋊10(C2×D4), (C2×C42)⋊9D5, C10⋊1(C4⋊1D4), (C4×C20)⋊50C22, (C22×D20)⋊4C2, C2.5(C22×D20), C10.3(C22×D4), (C2×D20)⋊42C22, (C2×C10).19C24, C22.64(C2×D20), (C2×C20).780C23, (C22×C4).438D10, (C22×D5).1C23, C22.62(C23×D5), (C23×D5).27C22, C23.315(C22×D5), (C22×C20).523C22, (C22×C10).381C23, (C2×C4×C20)⋊8C2, C5⋊1(C2×C4⋊1D4), (C2×C10).170(C2×D4), (C2×C4).729(C22×D5), SmallGroup(320,1147)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2430 in 498 conjugacy classes, 159 normal (9 characteristic)
C1, C2 [×7], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×40], C5, C2×C4 [×18], D4 [×48], C23, C23 [×32], D5 [×8], C10 [×7], C42 [×4], C22×C4 [×3], C2×D4 [×48], C24 [×4], C20 [×12], D10 [×40], C2×C10, C2×C10 [×6], C2×C42, C4⋊1D4 [×8], C22×D4 [×6], D20 [×48], C2×C20 [×18], C22×D5 [×8], C22×D5 [×24], C22×C10, C2×C4⋊1D4, C4×C20 [×4], C2×D20 [×24], C2×D20 [×24], C22×C20 [×3], C23×D5 [×4], C4⋊D20 [×8], C2×C4×C20, C22×D20 [×6], C2×C4⋊D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D5, C2×D4 [×18], C24, D10 [×7], C4⋊1D4 [×4], C22×D4 [×3], D20 [×12], C22×D5 [×7], C2×C4⋊1D4, C2×D20 [×18], C23×D5, C4⋊D20 [×4], C22×D20 [×3], C2×C4⋊D20
Generators and relations
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
►On 160 points
Generators in S160
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 130)(28 131)(29 132)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 139)(37 140)(38 121)(39 122)(40 123)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 101)(58 102)(59 103)(60 104)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)
(1 47 130 70)(2 48 131 71)(3 49 132 72)(4 50 133 73)(5 51 134 74)(6 52 135 75)(7 53 136 76)(8 54 137 77)(9 55 138 78)(10 56 139 79)(11 57 140 80)(12 58 121 61)(13 59 122 62)(14 60 123 63)(15 41 124 64)(16 42 125 65)(17 43 126 66)(18 44 127 67)(19 45 128 68)(20 46 129 69)(21 154 83 105)(22 155 84 106)(23 156 85 107)(24 157 86 108)(25 158 87 109)(26 159 88 110)(27 160 89 111)(28 141 90 112)(29 142 91 113)(30 143 92 114)(31 144 93 115)(32 145 94 116)(33 146 95 117)(34 147 96 118)(35 148 97 119)(36 149 98 120)(37 150 99 101)(38 151 100 102)(39 152 81 103)(40 153 82 104)
(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)
(1 149)(2 148)(3 147)(4 146)(5 145)(6 144)(7 143)(8 142)(9 141)(10 160)(11 159)(12 158)(13 157)(14 156)(15 155)(16 154)(17 153)(18 152)(19 151)(20 150)(21 42)(22 41)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 100)(69 99)(70 98)(71 97)(72 96)(73 95)(74 94)(75 93)(76 92)(77 91)(78 90)(79 89)(80 88)(101 129)(102 128)(103 127)(104 126)(105 125)(106 124)(107 123)(108 122)(109 121)(110 140)(111 139)(112 138)(113 137)(114 136)(115 135)(116 134)(117 133)(118 132)(119 131)(120 130)
G:=sub<Sym(160)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,130)(28,131)(29,132)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,140)(38,121)(39,122)(40,123)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,101)(58,102)(59,103)(60,104)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150), (1,47,130,70)(2,48,131,71)(3,49,132,72)(4,50,133,73)(5,51,134,74)(6,52,135,75)(7,53,136,76)(8,54,137,77)(9,55,138,78)(10,56,139,79)(11,57,140,80)(12,58,121,61)(13,59,122,62)(14,60,123,63)(15,41,124,64)(16,42,125,65)(17,43,126,66)(18,44,127,67)(19,45,128,68)(20,46,129,69)(21,154,83,105)(22,155,84,106)(23,156,85,107)(24,157,86,108)(25,158,87,109)(26,159,88,110)(27,160,89,111)(28,141,90,112)(29,142,91,113)(30,143,92,114)(31,144,93,115)(32,145,94,116)(33,146,95,117)(34,147,96,118)(35,148,97,119)(36,149,98,120)(37,150,99,101)(38,151,100,102)(39,152,81,103)(40,153,82,104), (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), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,42)(22,41)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122)(109,121)(110,140)(111,139)(112,138)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130)>;
G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,124)(22,125)(23,126)(24,127)(25,128)(26,129)(27,130)(28,131)(29,132)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,140)(38,121)(39,122)(40,123)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,101)(58,102)(59,103)(60,104)(61,151)(62,152)(63,153)(64,154)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150), (1,47,130,70)(2,48,131,71)(3,49,132,72)(4,50,133,73)(5,51,134,74)(6,52,135,75)(7,53,136,76)(8,54,137,77)(9,55,138,78)(10,56,139,79)(11,57,140,80)(12,58,121,61)(13,59,122,62)(14,60,123,63)(15,41,124,64)(16,42,125,65)(17,43,126,66)(18,44,127,67)(19,45,128,68)(20,46,129,69)(21,154,83,105)(22,155,84,106)(23,156,85,107)(24,157,86,108)(25,158,87,109)(26,159,88,110)(27,160,89,111)(28,141,90,112)(29,142,91,113)(30,143,92,114)(31,144,93,115)(32,145,94,116)(33,146,95,117)(34,147,96,118)(35,148,97,119)(36,149,98,120)(37,150,99,101)(38,151,100,102)(39,152,81,103)(40,153,82,104), (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), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,160)(11,159)(12,158)(13,157)(14,156)(15,155)(16,154)(17,153)(18,152)(19,151)(20,150)(21,42)(22,41)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,100)(69,99)(70,98)(71,97)(72,96)(73,95)(74,94)(75,93)(76,92)(77,91)(78,90)(79,89)(80,88)(101,129)(102,128)(103,127)(104,126)(105,125)(106,124)(107,123)(108,122)(109,121)(110,140)(111,139)(112,138)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130) );
G=PermutationGroup([(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,124),(22,125),(23,126),(24,127),(25,128),(26,129),(27,130),(28,131),(29,132),(30,133),(31,134),(32,135),(33,136),(34,137),(35,138),(36,139),(37,140),(38,121),(39,122),(40,123),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,101),(58,102),(59,103),(60,104),(61,151),(62,152),(63,153),(64,154),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,141),(72,142),(73,143),(74,144),(75,145),(76,146),(77,147),(78,148),(79,149),(80,150)], [(1,47,130,70),(2,48,131,71),(3,49,132,72),(4,50,133,73),(5,51,134,74),(6,52,135,75),(7,53,136,76),(8,54,137,77),(9,55,138,78),(10,56,139,79),(11,57,140,80),(12,58,121,61),(13,59,122,62),(14,60,123,63),(15,41,124,64),(16,42,125,65),(17,43,126,66),(18,44,127,67),(19,45,128,68),(20,46,129,69),(21,154,83,105),(22,155,84,106),(23,156,85,107),(24,157,86,108),(25,158,87,109),(26,159,88,110),(27,160,89,111),(28,141,90,112),(29,142,91,113),(30,143,92,114),(31,144,93,115),(32,145,94,116),(33,146,95,117),(34,147,96,118),(35,148,97,119),(36,149,98,120),(37,150,99,101),(38,151,100,102),(39,152,81,103),(40,153,82,104)], [(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)], [(1,149),(2,148),(3,147),(4,146),(5,145),(6,144),(7,143),(8,142),(9,141),(10,160),(11,159),(12,158),(13,157),(14,156),(15,155),(16,154),(17,153),(18,152),(19,151),(20,150),(21,42),(22,41),(23,60),(24,59),(25,58),(26,57),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,100),(69,99),(70,98),(71,97),(72,96),(73,95),(74,94),(75,93),(76,92),(77,91),(78,90),(79,89),(80,88),(101,129),(102,128),(103,127),(104,126),(105,125),(106,124),(107,123),(108,122),(109,121),(110,140),(111,139),(112,138),(113,137),(114,136),(115,135),(116,134),(117,133),(118,132),(119,131),(120,130)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 2 | 32 | 0 | 0 | 0 | 0 |
| 37 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 28 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 5 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 30 |
| 0 | 0 | 0 | 0 | 27 | 39 |
| 28 | 14 | 0 | 0 | 0 | 0 |
| 29 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 39 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 0 | 12 | 13 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[2,37,0,0,0,0,32,39,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,5,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,25,27,0,0,0,0,30,39],[28,29,0,0,0,0,14,13,0,0,0,0,0,0,16,25,0,0,0,0,39,25,0,0,0,0,0,0,28,12,0,0,0,0,27,13] >;
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | ··· | 20 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D20 |
| kernel | C2×C4⋊D20 | C4⋊D20 | C2×C4×C20 | C22×D20 | C2×C20 | C2×C42 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 8 | 1 | 6 | 12 | 2 | 8 | 6 | 48 |
In GAP, Magma, Sage, TeX
C_2\times C_4\rtimes D_{20} % in TeX
G:=Group("C2xC4:D20"); // GroupNames label
G:=SmallGroup(320,1147);
// by ID
G=gap.SmallGroup(320,1147);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀