C2^3:Dic10 - GroupNames

G = C₂³⋊Dic₁₀ order 320 = 2⁶·5

1^st semidirect product of C₂³ and Dic₁₀ acting via Dic₁₀/C₁₀=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₅D₁₀, C₂³⋊₁Dic₁₀, (C₂×C₂₀).₄₈D₄, (C₂²×C₁₀)⋊₃Q₈, C₅⋊₂(C₂³⋊Q₈), C₁₀.₃₆C₂²≀C₂, (C₂×Dic₅).₆₁D₄, (C₂²×C₄).₂₇D₁₀, C₂².₂₃₇(D₄×D₅), C₂.₆(C₂³⋊D₁₀), (C₂²×Dic₁₀)⋊₃C₂, C₁₀.₅₅(C₂²⋊Q₈), C₂.₆(C₂₀.₄₈D₄), C₁₀.₃₁(C₄.₄D₄), C₂.₅(C₂₀.₁₇D₄), C₂².₉₄(C₄○D₂₀), (C₂³×C₁₀).₃₀C₂², (C₂²×C₂₀).₅₆C₂², C₂².₄₄(C₂×Dic₁₀), C₂³.₃₆₆(C₂²×D₅), C₁₀.₁₀C₄²⋊₂₈C₂, C₂².₉₂(D₄⋊₂D₅), (C₂²×C₁₀).₃₂₂C₂³, C₂.₂₀(Dic₅.₅D₄), (C₂²×Dic₅).₃₈C₂², C₂.₂₀(Dic₅.₁₄D₄), (C₂×C₁₀).₃₂(C₂×Q₈), (C₂×C₁₀).₃₁₆(C₂×D₄), (C₂×C₄).₂₇(C₅⋊D₄), (C₂×C₂²⋊C₄).₁₀D₅, (C₂×C₂³.D₅).₉C₂, (C₂×C₁₀).₇₆(C₄○D₄), (C₁₀×C₂²⋊C₄).₁₂C₂, C₂².₁₂₂(C₂×C₅⋊D₄), SmallGroup(320,574)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — C₂³⋊Dic₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂²×Dic₅ — C₂²×Dic₁₀ — C₂³⋊Dic₁₀

Lower central

C₅ — C₂²×C₁₀ — C₂³⋊Dic₁₀

Upper central

C₁ — C₂³ — C₂×C₂²⋊C₄

Generators and relations for C₂³⋊Dic₁₀
G = < a,b,c,d,e | a²=b²=c²=d²⁰=1, e²=d¹⁰, eae^-1=ab=ba, dad^-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 710 in 202 conjugacy classes, 63 normal (27 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²×Q₈, Dic₁₀, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³⋊Q₈, C₂³.D₅, C₅×C₂²⋊C₄, C₂×Dic₁₀, C₂²×Dic₅, C₂²×C₂₀, C₂³×C₁₀, C₁₀.₁₀C₄², C₁₀.₁₀C₄², C₂×C₂³.D₅, C₁₀×C₂²⋊C₄, C₂²×Dic₁₀, C₂³⋊Dic₁₀
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₅, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₀, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, Dic₁₀, C₅⋊D₄, C₂²×D₅, C₂³⋊Q₈, C₂×Dic₁₀, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, C₂×C₅⋊D₄, Dic₅.₁₄D₄, Dic₅.₅D₄, C₂₀.₄₈D₄, C₂₀.₁₇D₄, C₂³⋊D₁₀, C₂³⋊Dic₁₀

Smallest permutation representation of C₂³⋊Dic₁₀
►On 160 points

Generators in S₁₆₀

(2 50)(4 52)(6 54)(8 56)(10 58)(12 60)(14 42)(16 44)(18 46)(20 48)(21 70)(23 72)(25 74)(27 76)(29 78)(31 80)(33 62)(35 64)(37 66)(39 68)(81 103)(82 154)(83 105)(84 156)(85 107)(86 158)(87 109)(88 160)(89 111)(90 142)(91 113)(92 144)(93 115)(94 146)(95 117)(96 148)(97 119)(98 150)(99 101)(100 152)(102 137)(104 139)(106 121)(108 123)(110 125)(112 127)(114 129)(116 131)(118 133)(120 135)(122 157)(124 159)(126 141)(128 143)(130 145)(132 147)(134 149)(136 151)(138 153)(140 155)

(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 61)(20 62)(21 56)(22 57)(23 58)(24 59)(25 60)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(81 103)(82 104)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 101)(100 102)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)

(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(81 138)(82 139)(83 140)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)

(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)

(1 126 11 136)(2 125 12 135)(3 124 13 134)(4 123 14 133)(5 122 15 132)(6 121 16 131)(7 140 17 130)(8 139 18 129)(9 138 19 128)(10 137 20 127)(21 104 31 114)(22 103 32 113)(23 102 33 112)(24 101 34 111)(25 120 35 110)(26 119 36 109)(27 118 37 108)(28 117 38 107)(29 116 39 106)(30 115 40 105)(41 97 51 87)(42 96 52 86)(43 95 53 85)(44 94 54 84)(45 93 55 83)(46 92 56 82)(47 91 57 81)(48 90 58 100)(49 89 59 99)(50 88 60 98)(61 143 71 153)(62 142 72 152)(63 141 73 151)(64 160 74 150)(65 159 75 149)(66 158 76 148)(67 157 77 147)(68 156 78 146)(69 155 79 145)(70 154 80 144)

G:=sub<Sym(160)| (2,50)(4,52)(6,54)(8,56)(10,58)(12,60)(14,42)(16,44)(18,46)(20,48)(21,70)(23,72)(25,74)(27,76)(29,78)(31,80)(33,62)(35,64)(37,66)(39,68)(81,103)(82,154)(83,105)(84,156)(85,107)(86,158)(87,109)(88,160)(89,111)(90,142)(91,113)(92,144)(93,115)(94,146)(95,117)(96,148)(97,119)(98,150)(99,101)(100,152)(102,137)(104,139)(106,121)(108,123)(110,125)(112,127)(114,129)(116,131)(118,133)(120,135)(122,157)(124,159)(126,141)(128,143)(130,145)(132,147)(134,149)(136,151)(138,153)(140,155), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,61)(20,62)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(81,103)(82,104)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,101)(100,102)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(81,138)(82,139)(83,140)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150), (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), (1,126,11,136)(2,125,12,135)(3,124,13,134)(4,123,14,133)(5,122,15,132)(6,121,16,131)(7,140,17,130)(8,139,18,129)(9,138,19,128)(10,137,20,127)(21,104,31,114)(22,103,32,113)(23,102,33,112)(24,101,34,111)(25,120,35,110)(26,119,36,109)(27,118,37,108)(28,117,38,107)(29,116,39,106)(30,115,40,105)(41,97,51,87)(42,96,52,86)(43,95,53,85)(44,94,54,84)(45,93,55,83)(46,92,56,82)(47,91,57,81)(48,90,58,100)(49,89,59,99)(50,88,60,98)(61,143,71,153)(62,142,72,152)(63,141,73,151)(64,160,74,150)(65,159,75,149)(66,158,76,148)(67,157,77,147)(68,156,78,146)(69,155,79,145)(70,154,80,144)>;

G:=Group( (2,50)(4,52)(6,54)(8,56)(10,58)(12,60)(14,42)(16,44)(18,46)(20,48)(21,70)(23,72)(25,74)(27,76)(29,78)(31,80)(33,62)(35,64)(37,66)(39,68)(81,103)(82,154)(83,105)(84,156)(85,107)(86,158)(87,109)(88,160)(89,111)(90,142)(91,113)(92,144)(93,115)(94,146)(95,117)(96,148)(97,119)(98,150)(99,101)(100,152)(102,137)(104,139)(106,121)(108,123)(110,125)(112,127)(114,129)(116,131)(118,133)(120,135)(122,157)(124,159)(126,141)(128,143)(130,145)(132,147)(134,149)(136,151)(138,153)(140,155), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,61)(20,62)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(81,103)(82,104)(83,105)(84,106)(85,107)(86,108)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,101)(100,102)(121,156)(122,157)(123,158)(124,159)(125,160)(126,141)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)(137,152)(138,153)(139,154)(140,155), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(81,138)(82,139)(83,140)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,129)(93,130)(94,131)(95,132)(96,133)(97,134)(98,135)(99,136)(100,137)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150), (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), (1,126,11,136)(2,125,12,135)(3,124,13,134)(4,123,14,133)(5,122,15,132)(6,121,16,131)(7,140,17,130)(8,139,18,129)(9,138,19,128)(10,137,20,127)(21,104,31,114)(22,103,32,113)(23,102,33,112)(24,101,34,111)(25,120,35,110)(26,119,36,109)(27,118,37,108)(28,117,38,107)(29,116,39,106)(30,115,40,105)(41,97,51,87)(42,96,52,86)(43,95,53,85)(44,94,54,84)(45,93,55,83)(46,92,56,82)(47,91,57,81)(48,90,58,100)(49,89,59,99)(50,88,60,98)(61,143,71,153)(62,142,72,152)(63,141,73,151)(64,160,74,150)(65,159,75,149)(66,158,76,148)(67,157,77,147)(68,156,78,146)(69,155,79,145)(70,154,80,144) );

G=PermutationGroup([[(2,50),(4,52),(6,54),(8,56),(10,58),(12,60),(14,42),(16,44),(18,46),(20,48),(21,70),(23,72),(25,74),(27,76),(29,78),(31,80),(33,62),(35,64),(37,66),(39,68),(81,103),(82,154),(83,105),(84,156),(85,107),(86,158),(87,109),(88,160),(89,111),(90,142),(91,113),(92,144),(93,115),(94,146),(95,117),(96,148),(97,119),(98,150),(99,101),(100,152),(102,137),(104,139),(106,121),(108,123),(110,125),(112,127),(114,129),(116,131),(118,133),(120,135),(122,157),(124,159),(126,141),(128,143),(130,145),(132,147),(134,149),(136,151),(138,153),(140,155)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,61),(20,62),(21,56),(22,57),(23,58),(24,59),(25,60),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(81,103),(82,104),(83,105),(84,106),(85,107),(86,108),(87,109),(88,110),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,101),(100,102),(121,156),(122,157),(123,158),(124,159),(125,160),(126,141),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151),(137,152),(138,153),(139,154),(140,155)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(81,138),(82,139),(83,140),(84,121),(85,122),(86,123),(87,124),(88,125),(89,126),(90,127),(91,128),(92,129),(93,130),(94,131),(95,132),(96,133),(97,134),(98,135),(99,136),(100,137),(101,151),(102,152),(103,153),(104,154),(105,155),(106,156),(107,157),(108,158),(109,159),(110,160),(111,141),(112,142),(113,143),(114,144),(115,145),(116,146),(117,147),(118,148),(119,149),(120,150)], [(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)], [(1,126,11,136),(2,125,12,135),(3,124,13,134),(4,123,14,133),(5,122,15,132),(6,121,16,131),(7,140,17,130),(8,139,18,129),(9,138,19,128),(10,137,20,127),(21,104,31,114),(22,103,32,113),(23,102,33,112),(24,101,34,111),(25,120,35,110),(26,119,36,109),(27,118,37,108),(28,117,38,107),(29,116,39,106),(30,115,40,105),(41,97,51,87),(42,96,52,86),(43,95,53,85),(44,94,54,84),(45,93,55,83),(46,92,56,82),(47,91,57,81),(48,90,58,100),(49,89,59,99),(50,88,60,98),(61,143,71,153),(62,142,72,152),(63,141,73,151),(64,160,74,150),(65,159,75,149),(66,158,76,148),(67,157,77,147),(68,156,78,146),(69,155,79,145),(70,154,80,144)]])

62 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	4B	4C	4D	4E	···	4L	5A	5B	10A	···	10N	10O	···	10V	20A	···	20P
order	1	2	···	2	2	2	4	4	4	4	4	···	4	5	5	10	···	10	10	···	10	20	···	20
size	1	1	···	1	4	4	4	4	4	4	20	···	20	2	2	2	···	2	4	···	4	4	···	4

62 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	-	+		+	+		-		+	-
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	Q₈	D₅	C₄○D₄	D₁₀	D₁₀	C₅⋊D₄	Dic₁₀	C₄○D₂₀	D₄×D₅	D₄⋊₂D₅
kernel	C₂³⋊Dic₁₀	C₁₀.₁₀C₄²	C₂×C₂³.D₅	C₁₀×C₂²⋊C₄	C₂²×Dic₁₀	C₂×Dic₅	C₂×C₂₀	C₂²×C₁₀	C₂×C₂²⋊C₄	C₂×C₁₀	C₂²×C₄	C₂⁴	C₂×C₄	C₂³	C₂²	C₂²	C₂²
# reps	1	3	2	1	1	4	2	2	2	6	4	2	8	8	8	4	4

Matrix representation of C₂³⋊Dic₁₀ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
40	1	0	0	0	0
0	0	1	0	0	0
0	0	1	40	0	0
0	0	0	0	1	0
0	0	0	0	26	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	39	0	0	0	0
0	40	0	0	0	0
0	0	1	39	0	0
0	0	0	40	0	0
0	0	0	0	21	0
0	0	0	0	1	2

40	2	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	26	39
0	0	0	0	31	15

G:=sub<GL(6,GF(41))| [40,40,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,26,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,1,0,0,0,0,0,39,40,0,0,0,0,0,0,21,1,0,0,0,0,0,2],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,31,0,0,0,0,39,15] >;

C₂³⋊Dic₁₀ in GAP, Magma, Sage, TeX

C_2^3\rtimes {\rm Dic}_{10}

% in TeX

G:=Group("C2^3:Dic10");

// GroupNames label

G:=SmallGroup(320,574);

// by ID

G=gap.SmallGroup(320,574);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,254,387,12550]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=d^10,e*a*e^-1=a*b=b*a,d*a*d^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C23⋊Dic10 order 320 = 26·5

1st semidirect product of C23 and Dic10 acting via Dic10/C10=C22

G = C₂³⋊Dic₁₀ order 320 = 2⁶·5

1^st semidirect product of C₂³ and Dic₁₀ acting via Dic₁₀/C₁₀=C₂²