?
G = C10.2+ (1+4) order 320 = 26·5
9th non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.42- (1+4), C10.92+ (1+4), (C2×C4)⋊4D20, (C2×C20)⋊9D4, C20⋊7D4⋊5C2, C4⋊2D20⋊9C2, C4.69(C2×D20), C4⋊C4.263D10, D10⋊2Q8⋊8C2, C20.222(C2×D4), C22.7(C2×D20), (C2×C10).54C24, C2.13(C22×D20), C10.11(C22×D4), (C2×C20).137C23, (C22×C4).179D10, C4⋊Dic5.29C22, C2.12(D4⋊6D10), C22.88(C23×D5), D10⋊C4.1C22, (C2×D20).262C22, (C22×C20).75C22, (C2×Dic5).16C23, (C22×D5).13C23, C23.226(C22×D5), (C22×C10).403C23, C2.7(Q8.10D10), C5⋊1(C22.31C24), (C2×Dic10).290C22, (C2×C4⋊C4)⋊19D5, (C10×C4⋊C4)⋊16C2, (C2×C4○D20)⋊16C2, (C2×C4×D5).64C22, (C2×C10).176(C2×D4), (C5×C4⋊C4).296C22, (C2×C4).572(C22×D5), (C2×C5⋊D4).101C22, SmallGroup(320,1182)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1182 in 294 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C5, C2×C4 [×10], C2×C4 [×14], D4 [×16], Q8 [×4], C23, C23 [×4], D5 [×4], C10 [×3], C10 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], C20 [×4], C20 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×8], D20 [×8], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×10], C2×C20 [×2], C22×D5 [×4], C22×C10, C22.31C24, C4⋊Dic5 [×4], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×6], C4○D20 [×8], C2×C5⋊D4 [×4], C22×C20, C22×C20 [×2], C4⋊2D20 [×4], D10⋊2Q8 [×4], C20⋊7D4 [×4], C10×C4⋊C4, C2×C4○D20 [×2], C10.2+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4), 2- (1+4), D20 [×4], C22×D5 [×7], C22.31C24, C2×D20 [×6], C23×D5, C22×D20, D4⋊6D10, Q8.10D10, C10.2+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, bab-1=dad-1=eae=a-1, ac=ca, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=ece=a5c, ede=a5b2d >
►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 70 30 56)(2 69 21 55)(3 68 22 54)(4 67 23 53)(5 66 24 52)(6 65 25 51)(7 64 26 60)(8 63 27 59)(9 62 28 58)(10 61 29 57)(11 130 151 116)(12 129 152 115)(13 128 153 114)(14 127 154 113)(15 126 155 112)(16 125 156 111)(17 124 157 120)(18 123 158 119)(19 122 159 118)(20 121 160 117)(31 76 45 90)(32 75 46 89)(33 74 47 88)(34 73 48 87)(35 72 49 86)(36 71 50 85)(37 80 41 84)(38 79 42 83)(39 78 43 82)(40 77 44 81)(91 131 105 145)(92 140 106 144)(93 139 107 143)(94 138 108 142)(95 137 109 141)(96 136 110 150)(97 135 101 149)(98 134 102 148)(99 133 103 147)(100 132 104 146)
(1 31 6 36)(2 32 7 37)(3 33 8 38)(4 34 9 39)(5 35 10 40)(11 131 16 136)(12 132 17 137)(13 133 18 138)(14 134 19 139)(15 135 20 140)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)(61 86 66 81)(62 87 67 82)(63 88 68 83)(64 89 69 84)(65 90 70 85)(91 130 96 125)(92 121 97 126)(93 122 98 127)(94 123 99 128)(95 124 100 129)(101 112 106 117)(102 113 107 118)(103 114 108 119)(104 115 109 120)(105 116 110 111)(141 152 146 157)(142 153 147 158)(143 154 148 159)(144 155 149 160)(145 156 150 151)
(1 16 30 156)(2 15 21 155)(3 14 22 154)(4 13 23 153)(5 12 24 152)(6 11 25 151)(7 20 26 160)(8 19 27 159)(9 18 28 158)(10 17 29 157)(31 131 45 145)(32 140 46 144)(33 139 47 143)(34 138 48 142)(35 137 49 141)(36 136 50 150)(37 135 41 149)(38 134 42 148)(39 133 43 147)(40 132 44 146)(51 111 65 125)(52 120 66 124)(53 119 67 123)(54 118 68 122)(55 117 69 121)(56 116 70 130)(57 115 61 129)(58 114 62 128)(59 113 63 127)(60 112 64 126)(71 105 85 91)(72 104 86 100)(73 103 87 99)(74 102 88 98)(75 101 89 97)(76 110 90 96)(77 109 81 95)(78 108 82 94)(79 107 83 93)(80 106 84 92)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 76)(12 75)(13 74)(14 73)(15 72)(16 71)(17 80)(18 79)(19 78)(20 77)(21 95)(22 94)(23 93)(24 92)(25 91)(26 100)(27 99)(28 98)(29 97)(30 96)(31 116)(32 115)(33 114)(34 113)(35 112)(36 111)(37 120)(38 119)(39 118)(40 117)(41 124)(42 123)(43 122)(44 121)(45 130)(46 129)(47 128)(48 127)(49 126)(50 125)(51 131)(52 140)(53 139)(54 138)(55 137)(56 136)(57 135)(58 134)(59 133)(60 132)(61 149)(62 148)(63 147)(64 146)(65 145)(66 144)(67 143)(68 142)(69 141)(70 150)(81 160)(82 159)(83 158)(84 157)(85 156)(86 155)(87 154)(88 153)(89 152)(90 151)
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), (1,70,30,56)(2,69,21,55)(3,68,22,54)(4,67,23,53)(5,66,24,52)(6,65,25,51)(7,64,26,60)(8,63,27,59)(9,62,28,58)(10,61,29,57)(11,130,151,116)(12,129,152,115)(13,128,153,114)(14,127,154,113)(15,126,155,112)(16,125,156,111)(17,124,157,120)(18,123,158,119)(19,122,159,118)(20,121,160,117)(31,76,45,90)(32,75,46,89)(33,74,47,88)(34,73,48,87)(35,72,49,86)(36,71,50,85)(37,80,41,84)(38,79,42,83)(39,78,43,82)(40,77,44,81)(91,131,105,145)(92,140,106,144)(93,139,107,143)(94,138,108,142)(95,137,109,141)(96,136,110,150)(97,135,101,149)(98,134,102,148)(99,133,103,147)(100,132,104,146), (1,31,6,36)(2,32,7,37)(3,33,8,38)(4,34,9,39)(5,35,10,40)(11,131,16,136)(12,132,17,137)(13,133,18,138)(14,134,19,139)(15,135,20,140)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75)(61,86,66,81)(62,87,67,82)(63,88,68,83)(64,89,69,84)(65,90,70,85)(91,130,96,125)(92,121,97,126)(93,122,98,127)(94,123,99,128)(95,124,100,129)(101,112,106,117)(102,113,107,118)(103,114,108,119)(104,115,109,120)(105,116,110,111)(141,152,146,157)(142,153,147,158)(143,154,148,159)(144,155,149,160)(145,156,150,151), (1,16,30,156)(2,15,21,155)(3,14,22,154)(4,13,23,153)(5,12,24,152)(6,11,25,151)(7,20,26,160)(8,19,27,159)(9,18,28,158)(10,17,29,157)(31,131,45,145)(32,140,46,144)(33,139,47,143)(34,138,48,142)(35,137,49,141)(36,136,50,150)(37,135,41,149)(38,134,42,148)(39,133,43,147)(40,132,44,146)(51,111,65,125)(52,120,66,124)(53,119,67,123)(54,118,68,122)(55,117,69,121)(56,116,70,130)(57,115,61,129)(58,114,62,128)(59,113,63,127)(60,112,64,126)(71,105,85,91)(72,104,86,100)(73,103,87,99)(74,102,88,98)(75,101,89,97)(76,110,90,96)(77,109,81,95)(78,108,82,94)(79,107,83,93)(80,106,84,92), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,80)(18,79)(19,78)(20,77)(21,95)(22,94)(23,93)(24,92)(25,91)(26,100)(27,99)(28,98)(29,97)(30,96)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,120)(38,119)(39,118)(40,117)(41,124)(42,123)(43,122)(44,121)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,131)(52,140)(53,139)(54,138)(55,137)(56,136)(57,135)(58,134)(59,133)(60,132)(61,149)(62,148)(63,147)(64,146)(65,145)(66,144)(67,143)(68,142)(69,141)(70,150)(81,160)(82,159)(83,158)(84,157)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)>;
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), (1,70,30,56)(2,69,21,55)(3,68,22,54)(4,67,23,53)(5,66,24,52)(6,65,25,51)(7,64,26,60)(8,63,27,59)(9,62,28,58)(10,61,29,57)(11,130,151,116)(12,129,152,115)(13,128,153,114)(14,127,154,113)(15,126,155,112)(16,125,156,111)(17,124,157,120)(18,123,158,119)(19,122,159,118)(20,121,160,117)(31,76,45,90)(32,75,46,89)(33,74,47,88)(34,73,48,87)(35,72,49,86)(36,71,50,85)(37,80,41,84)(38,79,42,83)(39,78,43,82)(40,77,44,81)(91,131,105,145)(92,140,106,144)(93,139,107,143)(94,138,108,142)(95,137,109,141)(96,136,110,150)(97,135,101,149)(98,134,102,148)(99,133,103,147)(100,132,104,146), (1,31,6,36)(2,32,7,37)(3,33,8,38)(4,34,9,39)(5,35,10,40)(11,131,16,136)(12,132,17,137)(13,133,18,138)(14,134,19,139)(15,135,20,140)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75)(61,86,66,81)(62,87,67,82)(63,88,68,83)(64,89,69,84)(65,90,70,85)(91,130,96,125)(92,121,97,126)(93,122,98,127)(94,123,99,128)(95,124,100,129)(101,112,106,117)(102,113,107,118)(103,114,108,119)(104,115,109,120)(105,116,110,111)(141,152,146,157)(142,153,147,158)(143,154,148,159)(144,155,149,160)(145,156,150,151), (1,16,30,156)(2,15,21,155)(3,14,22,154)(4,13,23,153)(5,12,24,152)(6,11,25,151)(7,20,26,160)(8,19,27,159)(9,18,28,158)(10,17,29,157)(31,131,45,145)(32,140,46,144)(33,139,47,143)(34,138,48,142)(35,137,49,141)(36,136,50,150)(37,135,41,149)(38,134,42,148)(39,133,43,147)(40,132,44,146)(51,111,65,125)(52,120,66,124)(53,119,67,123)(54,118,68,122)(55,117,69,121)(56,116,70,130)(57,115,61,129)(58,114,62,128)(59,113,63,127)(60,112,64,126)(71,105,85,91)(72,104,86,100)(73,103,87,99)(74,102,88,98)(75,101,89,97)(76,110,90,96)(77,109,81,95)(78,108,82,94)(79,107,83,93)(80,106,84,92), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,80)(18,79)(19,78)(20,77)(21,95)(22,94)(23,93)(24,92)(25,91)(26,100)(27,99)(28,98)(29,97)(30,96)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,120)(38,119)(39,118)(40,117)(41,124)(42,123)(43,122)(44,121)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,131)(52,140)(53,139)(54,138)(55,137)(56,136)(57,135)(58,134)(59,133)(60,132)(61,149)(62,148)(63,147)(64,146)(65,145)(66,144)(67,143)(68,142)(69,141)(70,150)(81,160)(82,159)(83,158)(84,157)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151) );
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)], [(1,70,30,56),(2,69,21,55),(3,68,22,54),(4,67,23,53),(5,66,24,52),(6,65,25,51),(7,64,26,60),(8,63,27,59),(9,62,28,58),(10,61,29,57),(11,130,151,116),(12,129,152,115),(13,128,153,114),(14,127,154,113),(15,126,155,112),(16,125,156,111),(17,124,157,120),(18,123,158,119),(19,122,159,118),(20,121,160,117),(31,76,45,90),(32,75,46,89),(33,74,47,88),(34,73,48,87),(35,72,49,86),(36,71,50,85),(37,80,41,84),(38,79,42,83),(39,78,43,82),(40,77,44,81),(91,131,105,145),(92,140,106,144),(93,139,107,143),(94,138,108,142),(95,137,109,141),(96,136,110,150),(97,135,101,149),(98,134,102,148),(99,133,103,147),(100,132,104,146)], [(1,31,6,36),(2,32,7,37),(3,33,8,38),(4,34,9,39),(5,35,10,40),(11,131,16,136),(12,132,17,137),(13,133,18,138),(14,134,19,139),(15,135,20,140),(21,46,26,41),(22,47,27,42),(23,48,28,43),(24,49,29,44),(25,50,30,45),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75),(61,86,66,81),(62,87,67,82),(63,88,68,83),(64,89,69,84),(65,90,70,85),(91,130,96,125),(92,121,97,126),(93,122,98,127),(94,123,99,128),(95,124,100,129),(101,112,106,117),(102,113,107,118),(103,114,108,119),(104,115,109,120),(105,116,110,111),(141,152,146,157),(142,153,147,158),(143,154,148,159),(144,155,149,160),(145,156,150,151)], [(1,16,30,156),(2,15,21,155),(3,14,22,154),(4,13,23,153),(5,12,24,152),(6,11,25,151),(7,20,26,160),(8,19,27,159),(9,18,28,158),(10,17,29,157),(31,131,45,145),(32,140,46,144),(33,139,47,143),(34,138,48,142),(35,137,49,141),(36,136,50,150),(37,135,41,149),(38,134,42,148),(39,133,43,147),(40,132,44,146),(51,111,65,125),(52,120,66,124),(53,119,67,123),(54,118,68,122),(55,117,69,121),(56,116,70,130),(57,115,61,129),(58,114,62,128),(59,113,63,127),(60,112,64,126),(71,105,85,91),(72,104,86,100),(73,103,87,99),(74,102,88,98),(75,101,89,97),(76,110,90,96),(77,109,81,95),(78,108,82,94),(79,107,83,93),(80,106,84,92)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,76),(12,75),(13,74),(14,73),(15,72),(16,71),(17,80),(18,79),(19,78),(20,77),(21,95),(22,94),(23,93),(24,92),(25,91),(26,100),(27,99),(28,98),(29,97),(30,96),(31,116),(32,115),(33,114),(34,113),(35,112),(36,111),(37,120),(38,119),(39,118),(40,117),(41,124),(42,123),(43,122),(44,121),(45,130),(46,129),(47,128),(48,127),(49,126),(50,125),(51,131),(52,140),(53,139),(54,138),(55,137),(56,136),(57,135),(58,134),(59,133),(60,132),(61,149),(62,148),(63,147),(64,146),(65,145),(66,144),(67,143),(68,142),(69,141),(70,150),(81,160),(82,159),(83,158),(84,157),(85,156),(86,155),(87,154),(88,153),(89,152),(90,151)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 34 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 38 | 0 | 0 |
| 0 | 0 | 38 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 38 |
| 0 | 0 | 0 | 0 | 38 | 20 |
| 20 | 36 | 0 | 0 | 0 | 0 |
| 31 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 20 | 36 | 0 | 0 | 0 | 0 |
| 6 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 24 | 18 | 21 |
| 0 | 0 | 18 | 38 | 26 | 23 |
| 0 | 0 | 18 | 21 | 38 | 17 |
| 0 | 0 | 26 | 23 | 23 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 32 | 10 | 22 |
| 0 | 0 | 32 | 19 | 22 | 31 |
| 0 | 0 | 10 | 22 | 19 | 9 |
| 0 | 0 | 22 | 31 | 9 | 22 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,0,0,0,0,35,6,0,0,0,0,34,0],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,21,38,0,0,0,0,38,20,0,0,0,0,0,0,21,38,0,0,0,0,38,20],[20,31,0,0,0,0,36,21,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[20,6,0,0,0,0,36,21,0,0,0,0,0,0,3,18,18,26,0,0,24,38,21,23,0,0,18,26,38,23,0,0,21,23,17,3],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,22,32,10,22,0,0,32,19,22,31,0,0,10,22,19,9,0,0,22,31,9,22] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D20 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | Q8.10D10 |
| kernel | C10.2+ (1+4) | C4⋊2D20 | D10⋊2Q8 | C20⋊7D4 | C10×C4⋊C4 | C2×C4○D20 | C2×C20 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 2 | 4 | 2 | 8 | 6 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}.2_+^{(1+4)} % in TeX
G:=Group("C10.ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1182);
// by ID
G=gap.SmallGroup(320,1182);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=b^2,b*a*b^-1=d*a*d^-1=e*a*e=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*b^2*d>;
// generators/relations
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