Copied to
clipboard

?

G = C4×Q8⋊2D5  order 320 = 26·5

Direct product of C4 and Q8⋊2D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×Q8⋊2D5, C42.231D10, Q8⋊8(C4×D5), (Q8×C20)⋊8C2, (C4×Q8)⋊19D5, D20⋊24(C2×C4), (C4×D20)⋊36C2, (D5×C42)⋊6C2, C20⋊18(C4○D4), C4⋊C4.324D10, (Q8×Dic5)⋊32C2, D20⋊8C4⋊47C2, C20.71(C22×C4), C10.47(C23×C4), (C2×Q8).201D10, Dic5⋊12(C4○D4), (C2×C10).117C24, (C2×C20).496C23, (C4×C20).169C22, D10.19(C22×C4), C22.36(C23×D5), (C2×D20).269C22, C4⋊Dic5.367C22, (Q8×C10).217C22, Dic5.55(C22×C4), (C2×Dic5).223C23, (C4×Dic5).283C22, (C22×D5).186C23, D10⋊C4.125C22, C5⋊5(C4×C4○D4), C4.36(C2×C4×D5), C2.6(D5×C4○D4), (C4×D5)⋊12(C2×C4), (C5×Q8)⋊20(C2×C4), C4⋊C4⋊7D5⋊47C2, C2.28(D5×C22×C4), C2.3(C2×Q8⋊2D5), C10.111(C2×C4○D4), (C2×C4×D5).316C22, (C5×C4⋊C4).345C22, (C2×Q8⋊2D5).15C2, (C2×C4).821(C22×D5), SmallGroup(320,1245)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C4×Q8⋊2D5
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8⋊2D5 — C4×Q8⋊2D5
Lower central
C5 — C10 — C4×Q8⋊2D5

Subgroups: 958 in 310 conjugacy classes, 157 normal (22 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×10], C22, C22 [×12], C5, C2×C4, C2×C4 [×6], C2×C4 [×29], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10 [×3], C42 [×3], C42 [×7], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×3], C20 [×8], C20 [×3], D10 [×6], D10 [×6], C2×C10, C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8, C4×Q8, C2×C4○D4, C4×D5 [×12], C4×D5 [×12], D20 [×12], C2×Dic5 [×2], C2×Dic5 [×3], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5 [×3], C4×C4○D4, C4×Dic5, C4×Dic5 [×6], C4⋊Dic5 [×3], D10⋊C4 [×6], C4×C20 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×9], C2×D20 [×3], Q8⋊2D5 [×8], Q8×C10, D5×C42 [×3], C4×D20 [×3], C4⋊C4⋊7D5 [×3], D20⋊8C4 [×3], Q8×Dic5, Q8×C20, C2×Q8⋊2D5, C4×Q8⋊2D5

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C4×C4○D4, C2×C4×D5 [×6], Q8⋊2D5 [×2], C23×D5, D5×C22×C4, C2×Q8⋊2D5, D5×C4○D4, C4×Q8⋊2D5

Generators and relations
 G = < a,b,c,d,e | a4=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Smallest permutation representation
►On 160 points
Generators in S160
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)(81 111 91 101)(82 112 92 102)(83 113 93 103)(84 114 94 104)(85 115 95 105)(86 116 96 106)(87 117 97 107)(88 118 98 108)(89 119 99 109)(90 120 100 110)(121 151 131 141)(122 152 132 142)(123 153 133 143)(124 154 134 144)(125 155 135 145)(126 156 136 146)(127 157 137 147)(128 158 138 148)(129 159 139 149)(130 160 140 150)

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

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

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

Matrix representation ►G ⊆ GL4(𝔽41) generated by

32000
03200
00320
00032
,
40000
04000
0001
00400
,
40000
04000
00032
00320
,
344000
1000
0010
0001
,
344000
7700
00040
00400
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,32,0,0,32,0],[34,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[34,7,0,0,40,7,0,0,0,0,0,40,0,0,40,0] >;
 

80 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4P4Q···4X4Y···4AD5A5B10A···10F20A···20H20I···20AF
order12222···244444···44···44···45510···1020···2020···20
size111110···1011112···25···510···10222···22···24···4

80 irreducible representations

dim111111111222222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C4D5C4○D4C4○D4D10D10D10C4×D5Q8⋊2D5D5×C4○D4
kernelC4×Q8⋊2D5D5×C42C4×D20C4⋊C4⋊7D5D20⋊8C4Q8×Dic5Q8×C20C2×Q8⋊2D5Q8⋊2D5C4×Q8Dic5C20C42C4⋊C4C2×Q8Q8C4C2
# reps13333111162446621644

In GAP, Magma, Sage, TeX 

C_4\times Q_8\rtimes_2D_5
 
% in TeX
 
G:=Group("C4xQ8:2D5");
 
// GroupNames label
 
G:=SmallGroup(320,1245);
 
// by ID
 
G=gap.SmallGroup(320,1245);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,184,80,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^5=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀