direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22.D4, (C23×C4)⋊4C10, (C23×C20)⋊7C2, C23.49(C5×D4), C24.32(C2×C10), C22.61(D4×C10), (C2×C20).657C23, (C2×C10).344C24, (C22×C20)⋊59C22, (C22×D4).10C10, C10.183(C22×D4), (C22×C10).171D4, C23.5(C22×C10), (D4×C10).316C22, (C23×C10).92C22, C22.18(C23×C10), (C22×C10).259C23, C2.7(D4×C2×C10), (C10×C4⋊C4)⋊43C2, (C2×C4⋊C4)⋊16C10, C4⋊C4⋊11(C2×C10), (D4×C2×C10).23C2, C2.7(C10×C4○D4), (C5×C4⋊C4)⋊67C22, (C2×C22⋊C4)⋊10C10, (C10×C22⋊C4)⋊30C2, C22⋊C4⋊12(C2×C10), (C22×C4)⋊17(C2×C10), (C2×D4).61(C2×C10), C10.226(C2×C4○D4), (C2×C10).415(C2×D4), C22.31(C5×C4○D4), (C5×C22⋊C4)⋊66C22, (C2×C4).13(C22×C10), (C2×C10).231(C4○D4), SmallGroup(320,1526)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 530 in 342 conjugacy classes, 178 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C5, C2×C4 [×10], C2×C4 [×18], D4 [×8], C23, C23 [×8], C23 [×10], C10, C10 [×6], C10 [×6], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×8], C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C20 [×10], C2×C10, C2×C10 [×10], C2×C10 [×22], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×C20 [×10], C2×C20 [×18], C5×D4 [×8], C22×C10, C22×C10 [×8], C22×C10 [×10], C2×C22.D4, C5×C22⋊C4 [×12], C5×C4⋊C4 [×8], C22×C20, C22×C20 [×8], C22×C20 [×4], D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C10×C22⋊C4, C10×C22⋊C4 [×2], C10×C4⋊C4 [×2], C5×C22.D4 [×8], C23×C20, D4×C2×C10, C10×C22.D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×4], C24, C2×C10 [×35], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C5×D4 [×4], C22×C10 [×15], C2×C22.D4, D4×C10 [×6], C5×C4○D4 [×4], C23×C10, C5×C22.D4 [×4], D4×C2×C10, C10×C4○D4 [×2], C10×C22.D4
Generators and relations
G = < a,b,c,d,e | a10=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
►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)
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 88)(12 89)(13 90)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 71)(29 72)(30 73)(31 97)(32 98)(33 99)(34 100)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 144)(42 145)(43 146)(44 147)(45 148)(46 149)(47 150)(48 141)(49 142)(50 143)(51 128)(52 129)(53 130)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 111)(69 112)(70 113)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 151)(109 152)(110 153)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 51)(7 52)(8 53)(9 54)(10 55)(11 40)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 159)(22 160)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(41 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 101)(80 102)(81 99)(82 100)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 141)(120 142)(121 139)(122 140)(123 131)(124 132)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)
(1 96 66 74)(2 97 67 75)(3 98 68 76)(4 99 69 77)(5 100 70 78)(6 91 61 79)(7 92 62 80)(8 93 63 71)(9 94 64 72)(10 95 65 73)(11 119 159 131)(12 120 160 132)(13 111 151 133)(14 112 152 134)(15 113 153 135)(16 114 154 136)(17 115 155 137)(18 116 156 138)(19 117 157 139)(20 118 158 140)(21 123 40 141)(22 124 31 142)(23 125 32 143)(24 126 33 144)(25 127 34 145)(26 128 35 146)(27 129 36 147)(28 130 37 148)(29 121 38 149)(30 122 39 150)(41 109 59 81)(42 110 60 82)(43 101 51 83)(44 102 52 84)(45 103 53 85)(46 104 54 86)(47 105 55 87)(48 106 56 88)(49 107 57 89)(50 108 58 90)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(71 108)(72 109)(73 110)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 91)(89 92)(90 93)(111 130)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)(131 146)(132 147)(133 148)(134 149)(135 150)(136 141)(137 142)(138 143)(139 144)(140 145)(151 156)(152 157)(153 158)(154 159)(155 160)
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), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,88)(12,89)(13,90)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,71)(29,72)(30,73)(31,97)(32,98)(33,99)(34,100)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,144)(42,145)(43,146)(44,147)(45,148)(46,149)(47,150)(48,141)(49,142)(50,143)(51,128)(52,129)(53,130)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,111)(69,112)(70,113)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,151)(109,152)(110,153), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(11,40)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,159)(22,160)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,101)(80,102)(81,99)(82,100)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,141)(120,142)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138), (1,96,66,74)(2,97,67,75)(3,98,68,76)(4,99,69,77)(5,100,70,78)(6,91,61,79)(7,92,62,80)(8,93,63,71)(9,94,64,72)(10,95,65,73)(11,119,159,131)(12,120,160,132)(13,111,151,133)(14,112,152,134)(15,113,153,135)(16,114,154,136)(17,115,155,137)(18,116,156,138)(19,117,157,139)(20,118,158,140)(21,123,40,141)(22,124,31,142)(23,125,32,143)(24,126,33,144)(25,127,34,145)(26,128,35,146)(27,129,36,147)(28,130,37,148)(29,121,38,149)(30,122,39,150)(41,109,59,81)(42,110,60,82)(43,101,51,83)(44,102,52,84)(45,103,53,85)(46,104,54,86)(47,105,55,87)(48,106,56,88)(49,107,57,89)(50,108,58,90), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(71,108)(72,109)(73,110)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,91)(89,92)(90,93)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,146)(132,147)(133,148)(134,149)(135,150)(136,141)(137,142)(138,143)(139,144)(140,145)(151,156)(152,157)(153,158)(154,159)(155,160)>;
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), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,88)(12,89)(13,90)(14,81)(15,82)(16,83)(17,84)(18,85)(19,86)(20,87)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,71)(29,72)(30,73)(31,97)(32,98)(33,99)(34,100)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,144)(42,145)(43,146)(44,147)(45,148)(46,149)(47,150)(48,141)(49,142)(50,143)(51,128)(52,129)(53,130)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,111)(69,112)(70,113)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,151)(109,152)(110,153), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(11,40)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,159)(22,160)(23,151)(24,152)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,101)(80,102)(81,99)(82,100)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,141)(120,142)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138), (1,96,66,74)(2,97,67,75)(3,98,68,76)(4,99,69,77)(5,100,70,78)(6,91,61,79)(7,92,62,80)(8,93,63,71)(9,94,64,72)(10,95,65,73)(11,119,159,131)(12,120,160,132)(13,111,151,133)(14,112,152,134)(15,113,153,135)(16,114,154,136)(17,115,155,137)(18,116,156,138)(19,117,157,139)(20,118,158,140)(21,123,40,141)(22,124,31,142)(23,125,32,143)(24,126,33,144)(25,127,34,145)(26,128,35,146)(27,129,36,147)(28,130,37,148)(29,121,38,149)(30,122,39,150)(41,109,59,81)(42,110,60,82)(43,101,51,83)(44,102,52,84)(45,103,53,85)(46,104,54,86)(47,105,55,87)(48,106,56,88)(49,107,57,89)(50,108,58,90), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(71,108)(72,109)(73,110)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,91)(89,92)(90,93)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,146)(132,147)(133,148)(134,149)(135,150)(136,141)(137,142)(138,143)(139,144)(140,145)(151,156)(152,157)(153,158)(154,159)(155,160) );
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)], [(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,88),(12,89),(13,90),(14,81),(15,82),(16,83),(17,84),(18,85),(19,86),(20,87),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,71),(29,72),(30,73),(31,97),(32,98),(33,99),(34,100),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,144),(42,145),(43,146),(44,147),(45,148),(46,149),(47,150),(48,141),(49,142),(50,143),(51,128),(52,129),(53,130),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,111),(69,112),(70,113),(101,154),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,151),(109,152),(110,153)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,51),(7,52),(8,53),(9,54),(10,55),(11,40),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,159),(22,160),(23,151),(24,152),(25,153),(26,154),(27,155),(28,156),(29,157),(30,158),(41,69),(42,70),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,101),(80,102),(81,99),(82,100),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(111,143),(112,144),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,141),(120,142),(121,139),(122,140),(123,131),(124,132),(125,133),(126,134),(127,135),(128,136),(129,137),(130,138)], [(1,96,66,74),(2,97,67,75),(3,98,68,76),(4,99,69,77),(5,100,70,78),(6,91,61,79),(7,92,62,80),(8,93,63,71),(9,94,64,72),(10,95,65,73),(11,119,159,131),(12,120,160,132),(13,111,151,133),(14,112,152,134),(15,113,153,135),(16,114,154,136),(17,115,155,137),(18,116,156,138),(19,117,157,139),(20,118,158,140),(21,123,40,141),(22,124,31,142),(23,125,32,143),(24,126,33,144),(25,127,34,145),(26,128,35,146),(27,129,36,147),(28,130,37,148),(29,121,38,149),(30,122,39,150),(41,109,59,81),(42,110,60,82),(43,101,51,83),(44,102,52,84),(45,103,53,85),(46,104,54,86),(47,105,55,87),(48,106,56,88),(49,107,57,89),(50,108,58,90)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,51),(49,52),(50,53),(71,108),(72,109),(73,110),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,91),(89,92),(90,93),(111,130),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,129),(131,146),(132,147),(133,148),(134,149),(135,150),(136,141),(137,142),(138,143),(139,144),(140,145),(151,156),(152,157),(153,158),(154,159),(155,160)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[40,0,0,0,0,0,0,32,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,40,0],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40] >;
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 10AS | ··· | 10AZ | 20A | ··· | 20AF | 20AG | ··· | 20BD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C10×C22.D4 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C22.D4 | C23×C20 | D4×C2×C10 | C2×C22.D4 | C2×C22⋊C4 | C2×C4⋊C4 | C22.D4 | C23×C4 | C22×D4 | C22×C10 | C2×C10 | C23 | C22 |
| # reps | 1 | 3 | 2 | 8 | 1 | 1 | 4 | 12 | 8 | 32 | 4 | 4 | 4 | 8 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times C_2^2.D_4 % in TeX
G:=Group("C10xC2^2.D4"); // GroupNames label
G:=SmallGroup(320,1526);
// by ID
G=gap.SmallGroup(320,1526);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,436]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations
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