?
G = C10.252- (1+4) order 320 = 26·5
25th non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.252- (1+4), C10.582+ (1+4), C20⋊Q8⋊29C2, C22⋊Q8⋊23D5, C4⋊C4.101D10, (C2×Q8).79D10, D10⋊Q8⋊24C2, D10⋊3Q8⋊25C2, C22⋊C4.23D10, C4.Dic10⋊27C2, Dic5⋊Q8⋊18C2, C20.48D4⋊47C2, (C2×C10).190C24, (C2×C20).176C23, (C22×C4).252D10, C2.60(D4⋊6D10), Dic5.Q8⋊22C2, D10.12D4.3C2, D10⋊C4.8C22, C23.D10⋊26C2, C4⋊Dic5.222C22, Dic5.5D4.3C2, (Q8×C10).119C22, (C2×Dic5).96C23, (C22×D5).81C23, C22.211(C23×D5), C23.126(C22×D5), C23.D5.36C22, (C22×C10).218C23, (C22×C20).318C22, C5⋊1(C22.57C24), (C2×Dic10).37C22, (C4×Dic5).125C22, C23.23D10.3C2, C10.D4.81C22, C2.39(D4.10D10), C2.26(Q8.10D10), C4⋊C4⋊D5⋊25C2, (C5×C22⋊Q8)⋊26C2, (C2×C4×D5).115C22, (C5×C4⋊C4).170C22, (C2×C4).187(C22×D5), (C2×C5⋊D4).42C22, (C5×C22⋊C4).45C22, SmallGroup(320,1318)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 646 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×9], D4, Q8 [×3], C23, C23, D5, C10 [×3], C10, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×7], C20 [×6], D10 [×3], C2×C10, C2×C10 [×3], C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic10 [×2], C4×D5, C2×Dic5 [×7], C5⋊D4, C2×C20 [×6], C2×C20, C5×Q8, C22×D5, C22×C10, C22.57C24, C4×Dic5 [×3], C10.D4 [×9], C4⋊Dic5 [×4], D10⋊C4 [×5], C23.D5 [×3], C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C2×Dic10 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, Q8×C10, C23.D10 [×2], D10.12D4, Dic5.5D4, C20⋊Q8, Dic5.Q8, C4.Dic10, D10⋊Q8, C4⋊C4⋊D5 [×2], C20.48D4, C23.23D10, Dic5⋊Q8, D10⋊3Q8, C5×C22⋊Q8, C10.252- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D5 [×7], C22.57C24, C23×D5, D4⋊6D10, Q8.10D10, D4.10D10, C10.252- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=a5b-1, bd=db, ebe-1=a5b, dcd-1=a5c, ce=ec, ede-1=b2d >
►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 45 30 33)(2 46 21 34)(3 47 22 35)(4 48 23 36)(5 49 24 37)(6 50 25 38)(7 41 26 39)(8 42 27 40)(9 43 28 31)(10 44 29 32)(11 138 153 150)(12 139 154 141)(13 140 155 142)(14 131 156 143)(15 132 157 144)(16 133 158 145)(17 134 159 146)(18 135 160 147)(19 136 151 148)(20 137 152 149)(51 88 63 76)(52 89 64 77)(53 90 65 78)(54 81 66 79)(55 82 67 80)(56 83 68 71)(57 84 69 72)(58 85 70 73)(59 86 61 74)(60 87 62 75)(91 116 103 128)(92 117 104 129)(93 118 105 130)(94 119 106 121)(95 120 107 122)(96 111 108 123)(97 112 109 124)(98 113 110 125)(99 114 101 126)(100 115 102 127)
(1 33 6 38)(2 32 7 37)(3 31 8 36)(4 40 9 35)(5 39 10 34)(11 145 16 150)(12 144 17 149)(13 143 18 148)(14 142 19 147)(15 141 20 146)(21 44 26 49)(22 43 27 48)(23 42 28 47)(24 41 29 46)(25 50 30 45)(51 82 56 87)(52 81 57 86)(53 90 58 85)(54 89 59 84)(55 88 60 83)(61 72 66 77)(62 71 67 76)(63 80 68 75)(64 79 69 74)(65 78 70 73)(91 122 96 127)(92 121 97 126)(93 130 98 125)(94 129 99 124)(95 128 100 123)(101 112 106 117)(102 111 107 116)(103 120 108 115)(104 119 109 114)(105 118 110 113)(131 160 136 155)(132 159 137 154)(133 158 138 153)(134 157 139 152)(135 156 140 151)
(1 85 30 73)(2 86 21 74)(3 87 22 75)(4 88 23 76)(5 89 24 77)(6 90 25 78)(7 81 26 79)(8 82 27 80)(9 83 28 71)(10 84 29 72)(11 93 153 105)(12 94 154 106)(13 95 155 107)(14 96 156 108)(15 97 157 109)(16 98 158 110)(17 99 159 101)(18 100 160 102)(19 91 151 103)(20 92 152 104)(31 56 43 68)(32 57 44 69)(33 58 45 70)(34 59 46 61)(35 60 47 62)(36 51 48 63)(37 52 49 64)(38 53 50 65)(39 54 41 66)(40 55 42 67)(111 143 123 131)(112 144 124 132)(113 145 125 133)(114 146 126 134)(115 147 127 135)(116 148 128 136)(117 149 129 137)(118 150 130 138)(119 141 121 139)(120 142 122 140)
(1 16 30 158)(2 15 21 157)(3 14 22 156)(4 13 23 155)(5 12 24 154)(6 11 25 153)(7 20 26 152)(8 19 27 151)(9 18 28 160)(10 17 29 159)(31 142 43 140)(32 141 44 139)(33 150 45 138)(34 149 46 137)(35 148 47 136)(36 147 48 135)(37 146 49 134)(38 145 50 133)(39 144 41 132)(40 143 42 131)(51 115 63 127)(52 114 64 126)(53 113 65 125)(54 112 66 124)(55 111 67 123)(56 120 68 122)(57 119 69 121)(58 118 70 130)(59 117 61 129)(60 116 62 128)(71 100 83 102)(72 99 84 101)(73 98 85 110)(74 97 86 109)(75 96 87 108)(76 95 88 107)(77 94 89 106)(78 93 90 105)(79 92 81 104)(80 91 82 103)
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), (1,45,30,33)(2,46,21,34)(3,47,22,35)(4,48,23,36)(5,49,24,37)(6,50,25,38)(7,41,26,39)(8,42,27,40)(9,43,28,31)(10,44,29,32)(11,138,153,150)(12,139,154,141)(13,140,155,142)(14,131,156,143)(15,132,157,144)(16,133,158,145)(17,134,159,146)(18,135,160,147)(19,136,151,148)(20,137,152,149)(51,88,63,76)(52,89,64,77)(53,90,65,78)(54,81,66,79)(55,82,67,80)(56,83,68,71)(57,84,69,72)(58,85,70,73)(59,86,61,74)(60,87,62,75)(91,116,103,128)(92,117,104,129)(93,118,105,130)(94,119,106,121)(95,120,107,122)(96,111,108,123)(97,112,109,124)(98,113,110,125)(99,114,101,126)(100,115,102,127), (1,33,6,38)(2,32,7,37)(3,31,8,36)(4,40,9,35)(5,39,10,34)(11,145,16,150)(12,144,17,149)(13,143,18,148)(14,142,19,147)(15,141,20,146)(21,44,26,49)(22,43,27,48)(23,42,28,47)(24,41,29,46)(25,50,30,45)(51,82,56,87)(52,81,57,86)(53,90,58,85)(54,89,59,84)(55,88,60,83)(61,72,66,77)(62,71,67,76)(63,80,68,75)(64,79,69,74)(65,78,70,73)(91,122,96,127)(92,121,97,126)(93,130,98,125)(94,129,99,124)(95,128,100,123)(101,112,106,117)(102,111,107,116)(103,120,108,115)(104,119,109,114)(105,118,110,113)(131,160,136,155)(132,159,137,154)(133,158,138,153)(134,157,139,152)(135,156,140,151), (1,85,30,73)(2,86,21,74)(3,87,22,75)(4,88,23,76)(5,89,24,77)(6,90,25,78)(7,81,26,79)(8,82,27,80)(9,83,28,71)(10,84,29,72)(11,93,153,105)(12,94,154,106)(13,95,155,107)(14,96,156,108)(15,97,157,109)(16,98,158,110)(17,99,159,101)(18,100,160,102)(19,91,151,103)(20,92,152,104)(31,56,43,68)(32,57,44,69)(33,58,45,70)(34,59,46,61)(35,60,47,62)(36,51,48,63)(37,52,49,64)(38,53,50,65)(39,54,41,66)(40,55,42,67)(111,143,123,131)(112,144,124,132)(113,145,125,133)(114,146,126,134)(115,147,127,135)(116,148,128,136)(117,149,129,137)(118,150,130,138)(119,141,121,139)(120,142,122,140), (1,16,30,158)(2,15,21,157)(3,14,22,156)(4,13,23,155)(5,12,24,154)(6,11,25,153)(7,20,26,152)(8,19,27,151)(9,18,28,160)(10,17,29,159)(31,142,43,140)(32,141,44,139)(33,150,45,138)(34,149,46,137)(35,148,47,136)(36,147,48,135)(37,146,49,134)(38,145,50,133)(39,144,41,132)(40,143,42,131)(51,115,63,127)(52,114,64,126)(53,113,65,125)(54,112,66,124)(55,111,67,123)(56,120,68,122)(57,119,69,121)(58,118,70,130)(59,117,61,129)(60,116,62,128)(71,100,83,102)(72,99,84,101)(73,98,85,110)(74,97,86,109)(75,96,87,108)(76,95,88,107)(77,94,89,106)(78,93,90,105)(79,92,81,104)(80,91,82,103)>;
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), (1,45,30,33)(2,46,21,34)(3,47,22,35)(4,48,23,36)(5,49,24,37)(6,50,25,38)(7,41,26,39)(8,42,27,40)(9,43,28,31)(10,44,29,32)(11,138,153,150)(12,139,154,141)(13,140,155,142)(14,131,156,143)(15,132,157,144)(16,133,158,145)(17,134,159,146)(18,135,160,147)(19,136,151,148)(20,137,152,149)(51,88,63,76)(52,89,64,77)(53,90,65,78)(54,81,66,79)(55,82,67,80)(56,83,68,71)(57,84,69,72)(58,85,70,73)(59,86,61,74)(60,87,62,75)(91,116,103,128)(92,117,104,129)(93,118,105,130)(94,119,106,121)(95,120,107,122)(96,111,108,123)(97,112,109,124)(98,113,110,125)(99,114,101,126)(100,115,102,127), (1,33,6,38)(2,32,7,37)(3,31,8,36)(4,40,9,35)(5,39,10,34)(11,145,16,150)(12,144,17,149)(13,143,18,148)(14,142,19,147)(15,141,20,146)(21,44,26,49)(22,43,27,48)(23,42,28,47)(24,41,29,46)(25,50,30,45)(51,82,56,87)(52,81,57,86)(53,90,58,85)(54,89,59,84)(55,88,60,83)(61,72,66,77)(62,71,67,76)(63,80,68,75)(64,79,69,74)(65,78,70,73)(91,122,96,127)(92,121,97,126)(93,130,98,125)(94,129,99,124)(95,128,100,123)(101,112,106,117)(102,111,107,116)(103,120,108,115)(104,119,109,114)(105,118,110,113)(131,160,136,155)(132,159,137,154)(133,158,138,153)(134,157,139,152)(135,156,140,151), (1,85,30,73)(2,86,21,74)(3,87,22,75)(4,88,23,76)(5,89,24,77)(6,90,25,78)(7,81,26,79)(8,82,27,80)(9,83,28,71)(10,84,29,72)(11,93,153,105)(12,94,154,106)(13,95,155,107)(14,96,156,108)(15,97,157,109)(16,98,158,110)(17,99,159,101)(18,100,160,102)(19,91,151,103)(20,92,152,104)(31,56,43,68)(32,57,44,69)(33,58,45,70)(34,59,46,61)(35,60,47,62)(36,51,48,63)(37,52,49,64)(38,53,50,65)(39,54,41,66)(40,55,42,67)(111,143,123,131)(112,144,124,132)(113,145,125,133)(114,146,126,134)(115,147,127,135)(116,148,128,136)(117,149,129,137)(118,150,130,138)(119,141,121,139)(120,142,122,140), (1,16,30,158)(2,15,21,157)(3,14,22,156)(4,13,23,155)(5,12,24,154)(6,11,25,153)(7,20,26,152)(8,19,27,151)(9,18,28,160)(10,17,29,159)(31,142,43,140)(32,141,44,139)(33,150,45,138)(34,149,46,137)(35,148,47,136)(36,147,48,135)(37,146,49,134)(38,145,50,133)(39,144,41,132)(40,143,42,131)(51,115,63,127)(52,114,64,126)(53,113,65,125)(54,112,66,124)(55,111,67,123)(56,120,68,122)(57,119,69,121)(58,118,70,130)(59,117,61,129)(60,116,62,128)(71,100,83,102)(72,99,84,101)(73,98,85,110)(74,97,86,109)(75,96,87,108)(76,95,88,107)(77,94,89,106)(78,93,90,105)(79,92,81,104)(80,91,82,103) );
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)], [(1,45,30,33),(2,46,21,34),(3,47,22,35),(4,48,23,36),(5,49,24,37),(6,50,25,38),(7,41,26,39),(8,42,27,40),(9,43,28,31),(10,44,29,32),(11,138,153,150),(12,139,154,141),(13,140,155,142),(14,131,156,143),(15,132,157,144),(16,133,158,145),(17,134,159,146),(18,135,160,147),(19,136,151,148),(20,137,152,149),(51,88,63,76),(52,89,64,77),(53,90,65,78),(54,81,66,79),(55,82,67,80),(56,83,68,71),(57,84,69,72),(58,85,70,73),(59,86,61,74),(60,87,62,75),(91,116,103,128),(92,117,104,129),(93,118,105,130),(94,119,106,121),(95,120,107,122),(96,111,108,123),(97,112,109,124),(98,113,110,125),(99,114,101,126),(100,115,102,127)], [(1,33,6,38),(2,32,7,37),(3,31,8,36),(4,40,9,35),(5,39,10,34),(11,145,16,150),(12,144,17,149),(13,143,18,148),(14,142,19,147),(15,141,20,146),(21,44,26,49),(22,43,27,48),(23,42,28,47),(24,41,29,46),(25,50,30,45),(51,82,56,87),(52,81,57,86),(53,90,58,85),(54,89,59,84),(55,88,60,83),(61,72,66,77),(62,71,67,76),(63,80,68,75),(64,79,69,74),(65,78,70,73),(91,122,96,127),(92,121,97,126),(93,130,98,125),(94,129,99,124),(95,128,100,123),(101,112,106,117),(102,111,107,116),(103,120,108,115),(104,119,109,114),(105,118,110,113),(131,160,136,155),(132,159,137,154),(133,158,138,153),(134,157,139,152),(135,156,140,151)], [(1,85,30,73),(2,86,21,74),(3,87,22,75),(4,88,23,76),(5,89,24,77),(6,90,25,78),(7,81,26,79),(8,82,27,80),(9,83,28,71),(10,84,29,72),(11,93,153,105),(12,94,154,106),(13,95,155,107),(14,96,156,108),(15,97,157,109),(16,98,158,110),(17,99,159,101),(18,100,160,102),(19,91,151,103),(20,92,152,104),(31,56,43,68),(32,57,44,69),(33,58,45,70),(34,59,46,61),(35,60,47,62),(36,51,48,63),(37,52,49,64),(38,53,50,65),(39,54,41,66),(40,55,42,67),(111,143,123,131),(112,144,124,132),(113,145,125,133),(114,146,126,134),(115,147,127,135),(116,148,128,136),(117,149,129,137),(118,150,130,138),(119,141,121,139),(120,142,122,140)], [(1,16,30,158),(2,15,21,157),(3,14,22,156),(4,13,23,155),(5,12,24,154),(6,11,25,153),(7,20,26,152),(8,19,27,151),(9,18,28,160),(10,17,29,159),(31,142,43,140),(32,141,44,139),(33,150,45,138),(34,149,46,137),(35,148,47,136),(36,147,48,135),(37,146,49,134),(38,145,50,133),(39,144,41,132),(40,143,42,131),(51,115,63,127),(52,114,64,126),(53,113,65,125),(54,112,66,124),(55,111,67,123),(56,120,68,122),(57,119,69,121),(58,118,70,130),(59,117,61,129),(60,116,62,128),(71,100,83,102),(72,99,84,101),(73,98,85,110),(74,97,86,109),(75,96,87,108),(76,95,88,107),(77,94,89,106),(78,93,90,105),(79,92,81,104),(80,91,82,103)])
Matrix representation ►G ⊆ GL10(𝔽41)
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 38 | 20 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 34 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 34 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 14 | 0 | 0 |
G:=sub<GL(10,GF(41))| [7,34,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0],[40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0],[40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0],[40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,21,0,0,3,0,0,0,0,0,0,0,21,38,0,0,0,0,0,0,0,0,38,20,0,0,0,0,0,0,0,3,0,0,20,0,0,0,0,0,0,0,0,0,0,0,0,27,34,0,0,0,0,0,0,0,0,34,14,0,0,0,0,0,0,27,34,0,0,0,0,0,0,0,0,34,14,0,0] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | Q8.10D10 | D4.10D10 |
| kernel | C10.252- (1+4) | C23.D10 | D10.12D4 | Dic5.5D4 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | D10⋊Q8 | C4⋊C4⋊D5 | C20.48D4 | C23.23D10 | Dic5⋊Q8 | D10⋊3Q8 | C5×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 6 | 2 | 2 | 1 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{25}2_-^{(1+4)} % in TeX
G:=Group("C10.25ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1318);
// by ID
G=gap.SmallGroup(320,1318);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=a^5,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b^-1,b*d=d*b,e*b*e^-1=a^5*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
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