direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4○D20, C10.4C25, D20⋊14C23, C20.77C24, D10.1C24, C24.73D10, Dic5.2C24, Dic10⋊13C23, (C23×C4)⋊8D5, (C4×D5)⋊8C23, C5⋊D4⋊7C23, C2.5(D5×C24), (C23×C20)⋊11C2, (C2×C20)⋊15C23, (C22×C4)⋊46D10, C4.76(C23×D5), (C22×D20)⋊25C2, (C2×D20)⋊66C22, C22.7(C23×D5), (C2×C10).326C24, (C22×C20)⋊62C22, (C22×Dic10)⋊26C2, (C2×Dic10)⋊77C22, C23.347(C22×D5), (C23×C10).116C22, (C22×C10).433C23, (C2×Dic5).306C23, (C22×D5).255C23, (C23×D5).128C22, (C22×Dic5).262C22, C10⋊1(C2×C4○D4), C5⋊1(C22×C4○D4), (C2×C4×D5)⋊72C22, (D5×C22×C4)⋊26C2, (C2×C4)⋊12(C22×D5), (C2×C10)⋊13(C4○D4), (C22×C5⋊D4)⋊22C2, (C2×C5⋊D4)⋊56C22, SmallGroup(320,1611)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C4○D20
G = < a,b,c,d,e | a2=b2=c4=e2=1, d10=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d9 >
Subgroups: 2558 in 890 conjugacy classes, 463 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C23×C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C22×C4○D4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C22×C20, C23×D5, C23×C10, C22×Dic10, D5×C22×C4, C22×D20, C2×C4○D20, C22×C5⋊D4, C23×C20, C22×C4○D20
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C25, C22×D5, C22×C4○D4, C4○D20, C23×D5, C2×C4○D20, D5×C24, C22×C4○D20
►On 160 points
Generators in S160
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 120)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 136)(72 137)(73 138)(74 139)(75 140)(76 121)(77 122)(78 123)(79 124)(80 125)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 159)(62 160)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 157)(80 158)(81 134)(82 135)(83 136)(84 137)(85 138)(86 139)(87 140)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 131)(99 132)(100 133)
(1 61 11 71)(2 62 12 72)(3 63 13 73)(4 64 14 74)(5 65 15 75)(6 66 16 76)(7 67 17 77)(8 68 18 78)(9 69 19 79)(10 70 20 80)(21 148 31 158)(22 149 32 159)(23 150 33 160)(24 151 34 141)(25 152 35 142)(26 153 36 143)(27 154 37 144)(28 155 38 145)(29 156 39 146)(30 157 40 147)(41 136 51 126)(42 137 52 127)(43 138 53 128)(44 139 54 129)(45 140 55 130)(46 121 56 131)(47 122 57 132)(48 123 58 133)(49 124 59 134)(50 125 60 135)(81 109 91 119)(82 110 92 120)(83 111 93 101)(84 112 94 102)(85 113 95 103)(86 114 96 104)(87 115 97 105)(88 116 98 106)(89 117 99 107)(90 118 100 108)
(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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 41)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(121 143)(122 142)(123 141)(124 160)(125 159)(126 158)(127 157)(128 156)(129 155)(130 154)(131 153)(132 152)(133 151)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)
G:=sub<Sym(160)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,120)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,121)(77,122)(78,123)(79,124)(80,125)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,159)(62,160)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133), (1,61,11,71)(2,62,12,72)(3,63,13,73)(4,64,14,74)(5,65,15,75)(6,66,16,76)(7,67,17,77)(8,68,18,78)(9,69,19,79)(10,70,20,80)(21,148,31,158)(22,149,32,159)(23,150,33,160)(24,151,34,141)(25,152,35,142)(26,153,36,143)(27,154,37,144)(28,155,38,145)(29,156,39,146)(30,157,40,147)(41,136,51,126)(42,137,52,127)(43,138,53,128)(44,139,54,129)(45,140,55,130)(46,121,56,131)(47,122,57,132)(48,123,58,133)(49,124,59,134)(50,125,60,135)(81,109,91,119)(82,110,92,120)(83,111,93,101)(84,112,94,102)(85,113,95,103)(86,114,96,104)(87,115,97,105)(88,116,98,106)(89,117,99,107)(90,118,100,108), (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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(121,143)(122,142)(123,141)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,120)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,121)(77,122)(78,123)(79,124)(80,125)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,159)(62,160)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133), (1,61,11,71)(2,62,12,72)(3,63,13,73)(4,64,14,74)(5,65,15,75)(6,66,16,76)(7,67,17,77)(8,68,18,78)(9,69,19,79)(10,70,20,80)(21,148,31,158)(22,149,32,159)(23,150,33,160)(24,151,34,141)(25,152,35,142)(26,153,36,143)(27,154,37,144)(28,155,38,145)(29,156,39,146)(30,157,40,147)(41,136,51,126)(42,137,52,127)(43,138,53,128)(44,139,54,129)(45,140,55,130)(46,121,56,131)(47,122,57,132)(48,123,58,133)(49,124,59,134)(50,125,60,135)(81,109,91,119)(82,110,92,120)(83,111,93,101)(84,112,94,102)(85,113,95,103)(86,114,96,104)(87,115,97,105)(88,116,98,106)(89,117,99,107)(90,118,100,108), (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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(121,143)(122,142)(123,141)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,120),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,136),(72,137),(73,138),(74,139),(75,140),(76,121),(77,122),(78,123),(79,124),(80,125),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,159),(62,160),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,155),(78,156),(79,157),(80,158),(81,134),(82,135),(83,136),(84,137),(85,138),(86,139),(87,140),(88,121),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(97,130),(98,131),(99,132),(100,133)], [(1,61,11,71),(2,62,12,72),(3,63,13,73),(4,64,14,74),(5,65,15,75),(6,66,16,76),(7,67,17,77),(8,68,18,78),(9,69,19,79),(10,70,20,80),(21,148,31,158),(22,149,32,159),(23,150,33,160),(24,151,34,141),(25,152,35,142),(26,153,36,143),(27,154,37,144),(28,155,38,145),(29,156,39,146),(30,157,40,147),(41,136,51,126),(42,137,52,127),(43,138,53,128),(44,139,54,129),(45,140,55,130),(46,121,56,131),(47,122,57,132),(48,123,58,133),(49,124,59,134),(50,125,60,135),(81,109,91,119),(82,110,92,120),(83,111,93,101),(84,112,94,102),(85,113,95,103),(86,114,96,104),(87,115,97,105),(88,116,98,106),(89,117,99,107),(90,118,100,108)], [(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)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,120),(12,119),(13,118),(14,117),(15,116),(16,115),(17,114),(18,113),(19,112),(20,111),(21,41),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(121,143),(122,142),(123,141),(124,160),(125,159),(126,158),(127,157),(128,156),(129,155),(130,154),(131,153),(132,152),(133,151),(134,150),(135,149),(136,148),(137,147),(138,146),(139,145),(140,144)]])
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | C4○D20 |
| kernel | C22×C4○D20 | C22×Dic10 | D5×C22×C4 | C22×D20 | C2×C4○D20 | C22×C5⋊D4 | C23×C20 | C23×C4 | C2×C10 | C22×C4 | C24 | C22 |
| # reps | 1 | 1 | 2 | 1 | 24 | 2 | 1 | 2 | 8 | 28 | 2 | 32 |
Matrix representation of C22×C4○D20 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,9,0,0,0,9,0,0,0,0,0,0,6,40,0,0,0,1,0],[1,0,0,0,0,0,0,32,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,35,1] >;
C22×C4○D20 in GAP, Magma, Sage, TeX
C_2^2\times C_4\circ D_{20} % in TeX
G:=Group("C2^2xC4oD20"); // GroupNames label
G:=SmallGroup(320,1611);
// by ID
G=gap.SmallGroup(320,1611);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,136,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^10=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^9>;
// generators/relations