?
G = C10.322+ (1+4) order 320 = 26·5
32nd non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.322+ (1+4), C10.682- (1+4), C4⋊D4⋊5D5, C4⋊C4.89D10, (C2×Dic5)⋊9D4, C20⋊7D4⋊43C2, C22.2(D4×D5), (C2×D4).89D10, C22⋊C4.4D10, D10⋊D4⋊16C2, D10⋊Q8⋊13C2, Dic5⋊D4⋊8C2, Dic5.85(C2×D4), C10.60(C22×D4), C23.8(C22×D5), (C2×C20).172C23, (C2×C10).141C24, (C2×D20).32C22, (C22×C4).217D10, C4⋊Dic5.44C22, C2.34(D4⋊6D10), (D4×C10).115C22, (C22×C10).12C23, (C22×D5).60C23, C22.162(C23×D5), Dic5.14D4⋊15C2, C23.D5.19C22, D10⋊C4.69C22, (C22×C20).310C22, C5⋊2(C22.31C24), (C2×Dic5).234C23, C2.26(D4.10D10), (C2×Dic10).157C22, C10.D4.158C22, (C22×Dic5).102C22, C2.33(C2×D4×D5), (C5×C4⋊D4)⋊6C2, (C2×C10).4(C2×D4), (C2×D4⋊2D5)⋊9C2, (C2×C4×D5).89C22, (C2×C4).35(C22×D5), (C2×C10.D4)⋊28C2, (C5×C4⋊C4).137C22, (C2×C5⋊D4).24C22, (C5×C22⋊C4).6C22, SmallGroup(320,1269)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1102 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20, C2×Dic5 [×10], C2×Dic5 [×5], C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20, C5×D4 [×5], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.31C24, C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, D4⋊2D5 [×8], C22×Dic5 [×2], C22×Dic5 [×2], C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], D10⋊D4 [×2], D10⋊Q8 [×2], C2×C10.D4, C20⋊7D4, Dic5⋊D4 [×4], C5×C4⋊D4, C2×D4⋊2D5 [×2], C10.322+ (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), C22×D5 [×7], C22.31C24, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊6D10, D4.10D10, C10.322+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, cac-1=eae=a-1, ad=da, 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)
(1 132 30 150)(2 133 21 141)(3 134 22 142)(4 135 23 143)(5 136 24 144)(6 137 25 145)(7 138 26 146)(8 139 27 147)(9 140 28 148)(10 131 29 149)(11 37 157 45)(12 38 158 46)(13 39 159 47)(14 40 160 48)(15 31 151 49)(16 32 152 50)(17 33 153 41)(18 34 154 42)(19 35 155 43)(20 36 156 44)(51 91 69 109)(52 92 70 110)(53 93 61 101)(54 94 62 102)(55 95 63 103)(56 96 64 104)(57 97 65 105)(58 98 66 106)(59 99 67 107)(60 100 68 108)(71 111 89 129)(72 112 90 130)(73 113 81 121)(74 114 82 122)(75 115 83 123)(76 116 84 124)(77 117 85 125)(78 118 86 126)(79 119 87 127)(80 120 88 128)
(1 45 6 50)(2 44 7 49)(3 43 8 48)(4 42 9 47)(5 41 10 46)(11 150 16 145)(12 149 17 144)(13 148 18 143)(14 147 19 142)(15 146 20 141)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)(51 73 56 78)(52 72 57 77)(53 71 58 76)(54 80 59 75)(55 79 60 74)(61 89 66 84)(62 88 67 83)(63 87 68 82)(64 86 69 81)(65 85 70 90)(91 126 96 121)(92 125 97 130)(93 124 98 129)(94 123 99 128)(95 122 100 127)(101 116 106 111)(102 115 107 120)(103 114 108 119)(104 113 109 118)(105 112 110 117)(131 153 136 158)(132 152 137 157)(133 151 138 156)(134 160 139 155)(135 159 140 154)
(1 125 30 117)(2 126 21 118)(3 127 22 119)(4 128 23 120)(5 129 24 111)(6 130 25 112)(7 121 26 113)(8 122 27 114)(9 123 28 115)(10 124 29 116)(11 65 157 57)(12 66 158 58)(13 67 159 59)(14 68 160 60)(15 69 151 51)(16 70 152 52)(17 61 153 53)(18 62 154 54)(19 63 155 55)(20 64 156 56)(31 104 49 96)(32 105 50 97)(33 106 41 98)(34 107 42 99)(35 108 43 100)(36 109 44 91)(37 110 45 92)(38 101 46 93)(39 102 47 94)(40 103 48 95)(71 149 89 131)(72 150 90 132)(73 141 81 133)(74 142 82 134)(75 143 83 135)(76 144 84 136)(77 145 85 137)(78 146 86 138)(79 147 87 139)(80 148 88 140)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 152)(12 151)(13 160)(14 159)(15 158)(16 157)(17 156)(18 155)(19 154)(20 153)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 50)(38 49)(39 48)(40 47)(51 53)(54 60)(55 59)(56 58)(61 69)(62 68)(63 67)(64 66)(71 78)(72 77)(73 76)(74 75)(79 80)(81 84)(82 83)(85 90)(86 89)(87 88)(91 93)(94 100)(95 99)(96 98)(101 109)(102 108)(103 107)(104 106)(111 118)(112 117)(113 116)(114 115)(119 120)(121 124)(122 123)(125 130)(126 129)(127 128)(131 141)(132 150)(133 149)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)
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), (1,132,30,150)(2,133,21,141)(3,134,22,142)(4,135,23,143)(5,136,24,144)(6,137,25,145)(7,138,26,146)(8,139,27,147)(9,140,28,148)(10,131,29,149)(11,37,157,45)(12,38,158,46)(13,39,159,47)(14,40,160,48)(15,31,151,49)(16,32,152,50)(17,33,153,41)(18,34,154,42)(19,35,155,43)(20,36,156,44)(51,91,69,109)(52,92,70,110)(53,93,61,101)(54,94,62,102)(55,95,63,103)(56,96,64,104)(57,97,65,105)(58,98,66,106)(59,99,67,107)(60,100,68,108)(71,111,89,129)(72,112,90,130)(73,113,81,121)(74,114,82,122)(75,115,83,123)(76,116,84,124)(77,117,85,125)(78,118,86,126)(79,119,87,127)(80,120,88,128), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,150,16,145)(12,149,17,144)(13,148,18,143)(14,147,19,142)(15,146,20,141)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37)(51,73,56,78)(52,72,57,77)(53,71,58,76)(54,80,59,75)(55,79,60,74)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,126,96,121)(92,125,97,130)(93,124,98,129)(94,123,99,128)(95,122,100,127)(101,116,106,111)(102,115,107,120)(103,114,108,119)(104,113,109,118)(105,112,110,117)(131,153,136,158)(132,152,137,157)(133,151,138,156)(134,160,139,155)(135,159,140,154), (1,125,30,117)(2,126,21,118)(3,127,22,119)(4,128,23,120)(5,129,24,111)(6,130,25,112)(7,121,26,113)(8,122,27,114)(9,123,28,115)(10,124,29,116)(11,65,157,57)(12,66,158,58)(13,67,159,59)(14,68,160,60)(15,69,151,51)(16,70,152,52)(17,61,153,53)(18,62,154,54)(19,63,155,55)(20,64,156,56)(31,104,49,96)(32,105,50,97)(33,106,41,98)(34,107,42,99)(35,108,43,100)(36,109,44,91)(37,110,45,92)(38,101,46,93)(39,102,47,94)(40,103,48,95)(71,149,89,131)(72,150,90,132)(73,141,81,133)(74,142,82,134)(75,143,83,135)(76,144,84,136)(77,145,85,137)(78,146,86,138)(79,147,87,139)(80,148,88,140), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,152)(12,151)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,50)(38,49)(39,48)(40,47)(51,53)(54,60)(55,59)(56,58)(61,69)(62,68)(63,67)(64,66)(71,78)(72,77)(73,76)(74,75)(79,80)(81,84)(82,83)(85,90)(86,89)(87,88)(91,93)(94,100)(95,99)(96,98)(101,109)(102,108)(103,107)(104,106)(111,118)(112,117)(113,116)(114,115)(119,120)(121,124)(122,123)(125,130)(126,129)(127,128)(131,141)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)>;
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), (1,132,30,150)(2,133,21,141)(3,134,22,142)(4,135,23,143)(5,136,24,144)(6,137,25,145)(7,138,26,146)(8,139,27,147)(9,140,28,148)(10,131,29,149)(11,37,157,45)(12,38,158,46)(13,39,159,47)(14,40,160,48)(15,31,151,49)(16,32,152,50)(17,33,153,41)(18,34,154,42)(19,35,155,43)(20,36,156,44)(51,91,69,109)(52,92,70,110)(53,93,61,101)(54,94,62,102)(55,95,63,103)(56,96,64,104)(57,97,65,105)(58,98,66,106)(59,99,67,107)(60,100,68,108)(71,111,89,129)(72,112,90,130)(73,113,81,121)(74,114,82,122)(75,115,83,123)(76,116,84,124)(77,117,85,125)(78,118,86,126)(79,119,87,127)(80,120,88,128), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,150,16,145)(12,149,17,144)(13,148,18,143)(14,147,19,142)(15,146,20,141)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37)(51,73,56,78)(52,72,57,77)(53,71,58,76)(54,80,59,75)(55,79,60,74)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,126,96,121)(92,125,97,130)(93,124,98,129)(94,123,99,128)(95,122,100,127)(101,116,106,111)(102,115,107,120)(103,114,108,119)(104,113,109,118)(105,112,110,117)(131,153,136,158)(132,152,137,157)(133,151,138,156)(134,160,139,155)(135,159,140,154), (1,125,30,117)(2,126,21,118)(3,127,22,119)(4,128,23,120)(5,129,24,111)(6,130,25,112)(7,121,26,113)(8,122,27,114)(9,123,28,115)(10,124,29,116)(11,65,157,57)(12,66,158,58)(13,67,159,59)(14,68,160,60)(15,69,151,51)(16,70,152,52)(17,61,153,53)(18,62,154,54)(19,63,155,55)(20,64,156,56)(31,104,49,96)(32,105,50,97)(33,106,41,98)(34,107,42,99)(35,108,43,100)(36,109,44,91)(37,110,45,92)(38,101,46,93)(39,102,47,94)(40,103,48,95)(71,149,89,131)(72,150,90,132)(73,141,81,133)(74,142,82,134)(75,143,83,135)(76,144,84,136)(77,145,85,137)(78,146,86,138)(79,147,87,139)(80,148,88,140), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,152)(12,151)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,50)(38,49)(39,48)(40,47)(51,53)(54,60)(55,59)(56,58)(61,69)(62,68)(63,67)(64,66)(71,78)(72,77)(73,76)(74,75)(79,80)(81,84)(82,83)(85,90)(86,89)(87,88)(91,93)(94,100)(95,99)(96,98)(101,109)(102,108)(103,107)(104,106)(111,118)(112,117)(113,116)(114,115)(119,120)(121,124)(122,123)(125,130)(126,129)(127,128)(131,141)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142) );
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)], [(1,132,30,150),(2,133,21,141),(3,134,22,142),(4,135,23,143),(5,136,24,144),(6,137,25,145),(7,138,26,146),(8,139,27,147),(9,140,28,148),(10,131,29,149),(11,37,157,45),(12,38,158,46),(13,39,159,47),(14,40,160,48),(15,31,151,49),(16,32,152,50),(17,33,153,41),(18,34,154,42),(19,35,155,43),(20,36,156,44),(51,91,69,109),(52,92,70,110),(53,93,61,101),(54,94,62,102),(55,95,63,103),(56,96,64,104),(57,97,65,105),(58,98,66,106),(59,99,67,107),(60,100,68,108),(71,111,89,129),(72,112,90,130),(73,113,81,121),(74,114,82,122),(75,115,83,123),(76,116,84,124),(77,117,85,125),(78,118,86,126),(79,119,87,127),(80,120,88,128)], [(1,45,6,50),(2,44,7,49),(3,43,8,48),(4,42,9,47),(5,41,10,46),(11,150,16,145),(12,149,17,144),(13,148,18,143),(14,147,19,142),(15,146,20,141),(21,36,26,31),(22,35,27,40),(23,34,28,39),(24,33,29,38),(25,32,30,37),(51,73,56,78),(52,72,57,77),(53,71,58,76),(54,80,59,75),(55,79,60,74),(61,89,66,84),(62,88,67,83),(63,87,68,82),(64,86,69,81),(65,85,70,90),(91,126,96,121),(92,125,97,130),(93,124,98,129),(94,123,99,128),(95,122,100,127),(101,116,106,111),(102,115,107,120),(103,114,108,119),(104,113,109,118),(105,112,110,117),(131,153,136,158),(132,152,137,157),(133,151,138,156),(134,160,139,155),(135,159,140,154)], [(1,125,30,117),(2,126,21,118),(3,127,22,119),(4,128,23,120),(5,129,24,111),(6,130,25,112),(7,121,26,113),(8,122,27,114),(9,123,28,115),(10,124,29,116),(11,65,157,57),(12,66,158,58),(13,67,159,59),(14,68,160,60),(15,69,151,51),(16,70,152,52),(17,61,153,53),(18,62,154,54),(19,63,155,55),(20,64,156,56),(31,104,49,96),(32,105,50,97),(33,106,41,98),(34,107,42,99),(35,108,43,100),(36,109,44,91),(37,110,45,92),(38,101,46,93),(39,102,47,94),(40,103,48,95),(71,149,89,131),(72,150,90,132),(73,141,81,133),(74,142,82,134),(75,143,83,135),(76,144,84,136),(77,145,85,137),(78,146,86,138),(79,147,87,139),(80,148,88,140)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,152),(12,151),(13,160),(14,159),(15,158),(16,157),(17,156),(18,155),(19,154),(20,153),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,50),(38,49),(39,48),(40,47),(51,53),(54,60),(55,59),(56,58),(61,69),(62,68),(63,67),(64,66),(71,78),(72,77),(73,76),(74,75),(79,80),(81,84),(82,83),(85,90),(86,89),(87,88),(91,93),(94,100),(95,99),(96,98),(101,109),(102,108),(103,107),(104,106),(111,118),(112,117),(113,116),(114,115),(119,120),(121,124),(122,123),(125,130),(126,129),(127,128),(131,141),(132,150),(133,149),(134,148),(135,147),(136,146),(137,145),(138,144),(139,143),(140,142)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 28 |
| 0 | 0 | 0 | 0 | 13 | 2 |
| 0 | 0 | 39 | 28 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 28 | 18 |
| 0 | 0 | 0 | 34 | 23 | 13 |
| 0 | 0 | 13 | 23 | 7 | 0 |
| 0 | 0 | 18 | 28 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,0,0,39,13,0,0,0,0,28,2,0,0,39,13,0,0,0,0,28,2,0,0],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,40,35,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,34,0,13,18,0,0,0,34,23,28,0,0,28,23,7,0,0,0,18,13,0,7],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
50 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 | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| 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 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4×D5 | D4⋊6D10 | D4.10D10 |
| kernel | C10.322+ (1+4) | Dic5.14D4 | D10⋊D4 | D10⋊Q8 | C2×C10.D4 | C20⋊7D4 | Dic5⋊D4 | C5×C4⋊D4 | C2×D4⋊2D5 | C2×Dic5 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 6 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{32}2_+^{(1+4)} % in TeX
G:=Group("C10.32ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1269);
// by ID
G=gap.SmallGroup(320,1269);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,1123,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*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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