?
G = C10.162- (1+4) order 320 = 26·5
16th non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.162- (1+4), C20⋊Q8⋊24C2, C22⋊Q8⋊5D5, C4⋊C4.95D10, (C4×D5).11D4, C4.187(D4×D5), D10.77(C2×D4), C20.232(C2×D4), D10⋊2Q8⋊24C2, D10⋊3Q8⋊14C2, (C2×C20).52C23, (C2×Q8).124D10, C22⋊C4.14D10, Dic5.19(C2×D4), C10.74(C22×D4), C20.48D4⋊36C2, (C2×C10).172C24, (C22×C4).234D10, D10.13D4⋊16C2, D10.12D4⋊24C2, Dic5.5D4⋊24C2, (C2×D20).229C22, C4⋊Dic5.213C22, (Q8×C10).105C22, (C2×Dic5).87C23, C22.193(C23×D5), C23.117(C22×D5), C23.D5.33C22, D10⋊C4.21C22, (C22×C10).200C23, (C22×C20).252C22, C5⋊2(C23.38C23), (C4×Dic5).112C22, C10.D4.25C22, (C22×D5).204C23, C2.35(D4.10D10), C2.17(Q8.10D10), (C2×Dic10).254C22, (C2×Q8×D5)⋊6C2, C2.47(C2×D4×D5), C4⋊C4⋊7D5⋊25C2, (C5×C22⋊Q8)⋊8C2, (C2×C4○D20).20C2, (C2×C4×D5).102C22, (C5×C4⋊C4).156C22, (C2×C4).590(C22×D5), (C2×C5⋊D4).128C22, (C5×C22⋊C4).27C22, SmallGroup(320,1300)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 958 in 270 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×6], Q8 [×10], C23, C23 [×2], D5 [×3], C10 [×3], C10, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×5], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C42⋊C2, C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×8], C4×D5 [×4], C4×D5 [×6], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10, C23.38C23, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×Dic10 [×2], C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, C4○D20 [×4], Q8×D5 [×4], C2×C5⋊D4 [×2], C22×C20, Q8×C10, D10.12D4 [×2], Dic5.5D4 [×2], C20⋊Q8 [×2], C4⋊C4⋊7D5, D10.13D4 [×2], D10⋊2Q8, C20.48D4, D10⋊3Q8, C5×C22⋊Q8, C2×C4○D20, C2×Q8×D5, C10.162- (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], C22×D5 [×7], C23.38C23, D4×D5 [×2], C23×D5, C2×D4×D5, Q8.10D10, D4.10D10, C10.162- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, 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)
(1 113 13 103)(2 114 14 104)(3 115 15 105)(4 116 16 106)(5 117 17 107)(6 118 18 108)(7 119 19 109)(8 120 20 110)(9 111 11 101)(10 112 12 102)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 156 51 146)(42 157 52 147)(43 158 53 148)(44 159 54 149)(45 160 55 150)(46 151 56 141)(47 152 57 142)(48 153 58 143)(49 154 59 144)(50 155 60 145)(61 136 71 126)(62 137 72 127)(63 138 73 128)(64 139 74 129)(65 140 75 130)(66 131 76 121)(67 132 77 122)(68 133 78 123)(69 134 79 124)(70 135 80 125)
(1 13)(2 12)(3 11)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(21 25)(22 24)(26 30)(27 29)(31 35)(32 34)(36 40)(37 39)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 95)(82 94)(83 93)(84 92)(85 91)(86 100)(87 99)(88 98)(89 97)(90 96)(101 105)(102 104)(106 110)(107 109)(111 115)(112 114)(116 120)(117 119)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)
(1 153 13 143)(2 152 14 142)(3 151 15 141)(4 160 16 150)(5 159 17 149)(6 158 18 148)(7 157 19 147)(8 156 20 146)(9 155 11 145)(10 154 12 144)(21 125 31 135)(22 124 32 134)(23 123 33 133)(24 122 34 132)(25 121 35 131)(26 130 36 140)(27 129 37 139)(28 128 38 138)(29 127 39 137)(30 126 40 136)(41 115 51 105)(42 114 52 104)(43 113 53 103)(44 112 54 102)(45 111 55 101)(46 120 56 110)(47 119 57 109)(48 118 58 108)(49 117 59 107)(50 116 60 106)(61 85 71 95)(62 84 72 94)(63 83 73 93)(64 82 74 92)(65 81 75 91)(66 90 76 100)(67 89 77 99)(68 88 78 98)(69 87 79 97)(70 86 80 96)
(1 93 13 83)(2 94 14 84)(3 95 15 85)(4 96 16 86)(5 97 17 87)(6 98 18 88)(7 99 19 89)(8 100 20 90)(9 91 11 81)(10 92 12 82)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(26 116 36 106)(27 117 37 107)(28 118 38 108)(29 119 39 109)(30 120 40 110)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)
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), (1,113,13,103)(2,114,14,104)(3,115,15,105)(4,116,16,106)(5,117,17,107)(6,118,18,108)(7,119,19,109)(8,120,20,110)(9,111,11,101)(10,112,12,102)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,156,51,146)(42,157,52,147)(43,158,53,148)(44,159,54,149)(45,160,55,150)(46,151,56,141)(47,152,57,142)(48,153,58,143)(49,154,59,144)(50,155,60,145)(61,136,71,126)(62,137,72,127)(63,138,73,128)(64,139,74,129)(65,140,75,130)(66,131,76,121)(67,132,77,122)(68,133,78,123)(69,134,79,124)(70,135,80,125), (1,13)(2,12)(3,11)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)(36,40)(37,39)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,95)(82,94)(83,93)(84,92)(85,91)(86,100)(87,99)(88,98)(89,97)(90,96)(101,105)(102,104)(106,110)(107,109)(111,115)(112,114)(116,120)(117,119)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151), (1,153,13,143)(2,152,14,142)(3,151,15,141)(4,160,16,150)(5,159,17,149)(6,158,18,148)(7,157,19,147)(8,156,20,146)(9,155,11,145)(10,154,12,144)(21,125,31,135)(22,124,32,134)(23,123,33,133)(24,122,34,132)(25,121,35,131)(26,130,36,140)(27,129,37,139)(28,128,38,138)(29,127,39,137)(30,126,40,136)(41,115,51,105)(42,114,52,104)(43,113,53,103)(44,112,54,102)(45,111,55,101)(46,120,56,110)(47,119,57,109)(48,118,58,108)(49,117,59,107)(50,116,60,106)(61,85,71,95)(62,84,72,94)(63,83,73,93)(64,82,74,92)(65,81,75,91)(66,90,76,100)(67,89,77,99)(68,88,78,98)(69,87,79,97)(70,86,80,96), (1,93,13,83)(2,94,14,84)(3,95,15,85)(4,96,16,86)(5,97,17,87)(6,98,18,88)(7,99,19,89)(8,100,20,90)(9,91,11,81)(10,92,12,82)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150)>;
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), (1,113,13,103)(2,114,14,104)(3,115,15,105)(4,116,16,106)(5,117,17,107)(6,118,18,108)(7,119,19,109)(8,120,20,110)(9,111,11,101)(10,112,12,102)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,156,51,146)(42,157,52,147)(43,158,53,148)(44,159,54,149)(45,160,55,150)(46,151,56,141)(47,152,57,142)(48,153,58,143)(49,154,59,144)(50,155,60,145)(61,136,71,126)(62,137,72,127)(63,138,73,128)(64,139,74,129)(65,140,75,130)(66,131,76,121)(67,132,77,122)(68,133,78,123)(69,134,79,124)(70,135,80,125), (1,13)(2,12)(3,11)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)(36,40)(37,39)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,95)(82,94)(83,93)(84,92)(85,91)(86,100)(87,99)(88,98)(89,97)(90,96)(101,105)(102,104)(106,110)(107,109)(111,115)(112,114)(116,120)(117,119)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151), (1,153,13,143)(2,152,14,142)(3,151,15,141)(4,160,16,150)(5,159,17,149)(6,158,18,148)(7,157,19,147)(8,156,20,146)(9,155,11,145)(10,154,12,144)(21,125,31,135)(22,124,32,134)(23,123,33,133)(24,122,34,132)(25,121,35,131)(26,130,36,140)(27,129,37,139)(28,128,38,138)(29,127,39,137)(30,126,40,136)(41,115,51,105)(42,114,52,104)(43,113,53,103)(44,112,54,102)(45,111,55,101)(46,120,56,110)(47,119,57,109)(48,118,58,108)(49,117,59,107)(50,116,60,106)(61,85,71,95)(62,84,72,94)(63,83,73,93)(64,82,74,92)(65,81,75,91)(66,90,76,100)(67,89,77,99)(68,88,78,98)(69,87,79,97)(70,86,80,96), (1,93,13,83)(2,94,14,84)(3,95,15,85)(4,96,16,86)(5,97,17,87)(6,98,18,88)(7,99,19,89)(8,100,20,90)(9,91,11,81)(10,92,12,82)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150) );
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)], [(1,113,13,103),(2,114,14,104),(3,115,15,105),(4,116,16,106),(5,117,17,107),(6,118,18,108),(7,119,19,109),(8,120,20,110),(9,111,11,101),(10,112,12,102),(21,91,31,81),(22,92,32,82),(23,93,33,83),(24,94,34,84),(25,95,35,85),(26,96,36,86),(27,97,37,87),(28,98,38,88),(29,99,39,89),(30,100,40,90),(41,156,51,146),(42,157,52,147),(43,158,53,148),(44,159,54,149),(45,160,55,150),(46,151,56,141),(47,152,57,142),(48,153,58,143),(49,154,59,144),(50,155,60,145),(61,136,71,126),(62,137,72,127),(63,138,73,128),(64,139,74,129),(65,140,75,130),(66,131,76,121),(67,132,77,122),(68,133,78,123),(69,134,79,124),(70,135,80,125)], [(1,13),(2,12),(3,11),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(21,25),(22,24),(26,30),(27,29),(31,35),(32,34),(36,40),(37,39),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,95),(82,94),(83,93),(84,92),(85,91),(86,100),(87,99),(88,98),(89,97),(90,96),(101,105),(102,104),(106,110),(107,109),(111,115),(112,114),(116,120),(117,119),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,160),(142,159),(143,158),(144,157),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151)], [(1,153,13,143),(2,152,14,142),(3,151,15,141),(4,160,16,150),(5,159,17,149),(6,158,18,148),(7,157,19,147),(8,156,20,146),(9,155,11,145),(10,154,12,144),(21,125,31,135),(22,124,32,134),(23,123,33,133),(24,122,34,132),(25,121,35,131),(26,130,36,140),(27,129,37,139),(28,128,38,138),(29,127,39,137),(30,126,40,136),(41,115,51,105),(42,114,52,104),(43,113,53,103),(44,112,54,102),(45,111,55,101),(46,120,56,110),(47,119,57,109),(48,118,58,108),(49,117,59,107),(50,116,60,106),(61,85,71,95),(62,84,72,94),(63,83,73,93),(64,82,74,92),(65,81,75,91),(66,90,76,100),(67,89,77,99),(68,88,78,98),(69,87,79,97),(70,86,80,96)], [(1,93,13,83),(2,94,14,84),(3,95,15,85),(4,96,16,86),(5,97,17,87),(6,98,18,88),(7,99,19,89),(8,100,20,90),(9,91,11,81),(10,92,12,82),(21,111,31,101),(22,112,32,102),(23,113,33,103),(24,114,34,104),(25,115,35,105),(26,116,36,106),(27,117,37,107),(28,118,38,108),(29,119,39,109),(30,120,40,110),(41,131,51,121),(42,132,52,122),(43,133,53,123),(44,134,54,124),(45,135,55,125),(46,136,56,126),(47,137,57,127),(48,138,58,128),(49,139,59,129),(50,140,60,130),(61,151,71,141),(62,152,72,142),(63,153,73,143),(64,154,74,144),(65,155,75,145),(66,156,76,146),(67,157,77,147),(68,158,78,148),(69,159,79,149),(70,160,80,150)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 35 | 5 | 0 | 0 | 0 | 0 |
| 34 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 25 | 38 | 0 |
| 0 | 0 | 7 | 15 | 21 | 3 |
| 0 | 0 | 38 | 0 | 26 | 25 |
| 0 | 0 | 21 | 3 | 7 | 15 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[1,27,0,0,0,0,0,40,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,11,32,0,0,0,0,9,30],[1,27,0,0,0,0,0,40,0,0,0,0,0,0,40,7,0,0,0,0,0,1,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[35,34,0,0,0,0,5,6,0,0,0,0,0,0,26,7,38,21,0,0,25,15,0,3,0,0,38,21,26,7,0,0,0,3,25,15],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,11,32,0,0,0,0,9,30,0,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | Q8.10D10 | D4.10D10 |
| kernel | C10.162- (1+4) | D10.12D4 | Dic5.5D4 | C20⋊Q8 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊2Q8 | C20.48D4 | D10⋊3Q8 | C5×C22⋊Q8 | C2×C4○D20 | C2×Q8×D5 | C4×D5 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 2 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{16}2_-^{(1+4)} % in TeX
G:=Group("C10.16ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1300);
// by ID
G=gap.SmallGroup(320,1300);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
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