metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2xC4):9D20, (C2xC20):19D4, C2.5(C4xD20), (C2xD20):16C4, C10.32(C4xD4), C10.2C22wrC2, (C2xDic5):15D4, C22.59(D4xD5), D10:2(C22:C4), C10.3(C4:D4), C2.1(C4:D20), (C22xD20).1C2, C2.C42:7D5, (C22xC4).15D10, C22.24(C2xD20), C2.1(C22:D20), C2.2(D10:D4), C2.5(D20:8C4), (C22xD5).103D4, C5:2(C23.23D4), C22.34(C4oD20), (C23xD5).94C22, C23.255(C22xD5), C10.10C42:27C2, C2.3(D10.13D4), (C22xC10).290C23, (C22xC20).331C22, C22.17(Q8:2D5), C10.38(C22.D4), (C22xDic5).14C22, (C2xC4):2(C4xD5), (C2xC20):17(C2xC4), (D5xC22xC4):12C2, C2.7(D5xC22:C4), C22.89(C2xC4xD5), (C22xD5):5(C2xC4), (C2xC10).199(C2xD4), C10.45(C2xC22:C4), (C2xD10:C4):28C2, (C2xC10).183(C4oD4), (C5xC2.C42):14C2, (C2xC10).150(C22xC4), SmallGroup(320,292)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2xC4):9D20
G = < a,b,c,d | a2=b4=c20=d2=1, cbc-1=dbd=ab=ba, ac=ca, ad=da, dcd=c-1 >
Subgroups: 1294 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2xC4, C2xC4, D4, C23, C23, D5, C10, C22:C4, C22xC4, C22xC4, C2xD4, C24, Dic5, C20, D10, D10, C2xC10, C2.C42, C2.C42, C2xC22:C4, C23xC4, C22xD4, C4xD5, D20, C2xDic5, C2xDic5, C2xC20, C2xC20, C22xD5, C22xD5, C22xC10, C23.23D4, D10:C4, C2xC4xD5, C2xD20, C2xD20, C22xDic5, C22xC20, C23xD5, C10.10C42, C5xC2.C42, C2xD10:C4, D5xC22xC4, C22xD20, (C2xC4):9D20
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22:C4, C22xC4, C2xD4, C4oD4, D10, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22.D4, C4xD5, D20, C22xD5, C23.23D4, C2xC4xD5, C2xD20, C4oD20, D4xD5, Q8:2D5, C4xD20, D5xC22:C4, C22:D20, D10:D4, D20:8C4, D10.13D4, C4:D20, (C2xC4):9D20
►On 160 points
Generators in S160
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 101)(40 102)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(73 159)(74 160)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)
(1 76 105 47)(2 143 106 98)(3 78 107 49)(4 145 108 100)(5 80 109 51)(6 147 110 82)(7 62 111 53)(8 149 112 84)(9 64 113 55)(10 151 114 86)(11 66 115 57)(12 153 116 88)(13 68 117 59)(14 155 118 90)(15 70 119 41)(16 157 120 92)(17 72 101 43)(18 159 102 94)(19 74 103 45)(20 141 104 96)(21 95 127 160)(22 46 128 75)(23 97 129 142)(24 48 130 77)(25 99 131 144)(26 50 132 79)(27 81 133 146)(28 52 134 61)(29 83 135 148)(30 54 136 63)(31 85 137 150)(32 56 138 65)(33 87 139 152)(34 58 140 67)(35 89 121 154)(36 60 122 69)(37 91 123 156)(38 42 124 71)(39 93 125 158)(40 44 126 73)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 97)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 100)(59 99)(60 98)(61 151)(62 150)(63 149)(64 148)(65 147)(66 146)(67 145)(68 144)(69 143)(70 142)(71 141)(72 160)(73 159)(74 158)(75 157)(76 156)(77 155)(78 154)(79 153)(80 152)(101 103)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(121 131)(122 130)(123 129)(124 128)(125 127)(132 140)(133 139)(134 138)(135 137)
G:=sub<Sym(160)| (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(73,159)(74,160)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146), (1,76,105,47)(2,143,106,98)(3,78,107,49)(4,145,108,100)(5,80,109,51)(6,147,110,82)(7,62,111,53)(8,149,112,84)(9,64,113,55)(10,151,114,86)(11,66,115,57)(12,153,116,88)(13,68,117,59)(14,155,118,90)(15,70,119,41)(16,157,120,92)(17,72,101,43)(18,159,102,94)(19,74,103,45)(20,141,104,96)(21,95,127,160)(22,46,128,75)(23,97,129,142)(24,48,130,77)(25,99,131,144)(26,50,132,79)(27,81,133,146)(28,52,134,61)(29,83,135,148)(30,54,136,63)(31,85,137,150)(32,56,138,65)(33,87,139,152)(34,58,140,67)(35,89,121,154)(36,60,122,69)(37,91,123,156)(38,42,124,71)(39,93,125,158)(40,44,126,73), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,100)(59,99)(60,98)(61,151)(62,150)(63,149)(64,148)(65,147)(66,146)(67,145)(68,144)(69,143)(70,142)(71,141)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,131)(122,130)(123,129)(124,128)(125,127)(132,140)(133,139)(134,138)(135,137)>;
G:=Group( (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(73,159)(74,160)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146), (1,76,105,47)(2,143,106,98)(3,78,107,49)(4,145,108,100)(5,80,109,51)(6,147,110,82)(7,62,111,53)(8,149,112,84)(9,64,113,55)(10,151,114,86)(11,66,115,57)(12,153,116,88)(13,68,117,59)(14,155,118,90)(15,70,119,41)(16,157,120,92)(17,72,101,43)(18,159,102,94)(19,74,103,45)(20,141,104,96)(21,95,127,160)(22,46,128,75)(23,97,129,142)(24,48,130,77)(25,99,131,144)(26,50,132,79)(27,81,133,146)(28,52,134,61)(29,83,135,148)(30,54,136,63)(31,85,137,150)(32,56,138,65)(33,87,139,152)(34,58,140,67)(35,89,121,154)(36,60,122,69)(37,91,123,156)(38,42,124,71)(39,93,125,158)(40,44,126,73), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,97)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,100)(59,99)(60,98)(61,151)(62,150)(63,149)(64,148)(65,147)(66,146)(67,145)(68,144)(69,143)(70,142)(71,141)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(101,103)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(121,131)(122,130)(123,129)(124,128)(125,127)(132,140)(133,139)(134,138)(135,137) );
G=PermutationGroup([[(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,136),(9,137),(10,138),(11,139),(12,140),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,101),(40,102),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,147),(62,148),(63,149),(64,150),(65,151),(66,152),(67,153),(68,154),(69,155),(70,156),(71,157),(72,158),(73,159),(74,160),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146)], [(1,76,105,47),(2,143,106,98),(3,78,107,49),(4,145,108,100),(5,80,109,51),(6,147,110,82),(7,62,111,53),(8,149,112,84),(9,64,113,55),(10,151,114,86),(11,66,115,57),(12,153,116,88),(13,68,117,59),(14,155,118,90),(15,70,119,41),(16,157,120,92),(17,72,101,43),(18,159,102,94),(19,74,103,45),(20,141,104,96),(21,95,127,160),(22,46,128,75),(23,97,129,142),(24,48,130,77),(25,99,131,144),(26,50,132,79),(27,81,133,146),(28,52,134,61),(29,83,135,148),(30,54,136,63),(31,85,137,150),(32,56,138,65),(33,87,139,152),(34,58,140,67),(35,89,121,154),(36,60,122,69),(37,91,123,156),(38,42,124,71),(39,93,125,158),(40,44,126,73)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,97),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,100),(59,99),(60,98),(61,151),(62,150),(63,149),(64,148),(65,147),(66,146),(67,145),(68,144),(69,143),(70,142),(71,141),(72,160),(73,159),(74,158),(75,157),(76,156),(77,155),(78,154),(79,153),(80,152),(101,103),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(121,131),(122,130),(123,129),(124,128),(125,127),(132,140),(133,139),(134,138),(135,137)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | C4oD4 | D10 | C4xD5 | D20 | C4oD20 | D4xD5 | Q8:2D5 |
| kernel | (C2xC4):9D20 | C10.10C42 | C5xC2.C42 | C2xD10:C4 | D5xC22xC4 | C22xD20 | C2xD20 | C2xDic5 | C2xC20 | C22xD5 | C2.C42 | C2xC10 | C22xC4 | C2xC4 | C2xC4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 3 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 4 | 6 | 8 | 8 | 8 | 6 | 2 |
Matrix representation of (C2xC4):9D20 ►in GL6(F41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 30 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,30,0,0,0,0,11,14,0,0,0,0,0,0,34,8,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
(C2xC4):9D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_9D_{20} % in TeX
G:=Group("(C2xC4):9D20"); // GroupNames label
G:=SmallGroup(320,292);
// by ID
G=gap.SmallGroup(320,292);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,c*b*c^-1=d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,d*c*d=c^-1>;
// generators/relations