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G = C4×Q8⋊D5order 320 = 26·5

Direct product of C4 and Q8⋊D5

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

Aliases: C4×Q8⋊D5, C2014SD16, C42.211D10, (C4×Q8)⋊1D5, Q83(C4×D5), C57(C4×SD16), (Q8×C20)⋊1C2, C4⋊C4.251D10, D20.30(C2×C4), (C4×D20).14C2, (C2×C20).257D4, C10.103(C4×D4), C20.58(C4○D4), C4.40(C4○D20), C10.92(C4○D8), Q8⋊Dic543C2, C20.60(C22×C4), (C4×C20).96C22, (C2×Q8).158D10, C20.Q845C2, C10.69(C2×SD16), D206C4.17C2, (C2×C20).345C23, C2.5(D4.8D10), (C2×D20).246C22, C4⋊Dic5.330C22, (Q8×C10).193C22, C4.25(C2×C4×D5), (C4×C52C8)⋊10C2, C2.3(C2×Q8⋊D5), C52C822(C2×C4), (C5×Q8)⋊17(C2×C4), C2.19(C4×C5⋊D4), (C2×Q8⋊D5).10C2, (C2×C10).476(C2×D4), C22.79(C2×C5⋊D4), (C2×C4).103(C5⋊D4), (C5×C4⋊C4).282C22, (C2×C4).445(C22×D5), (C2×C52C8).254C22, SmallGroup(320,652)

Series: Derived Chief Lower central Upper central

C1C20 — C4×Q8⋊D5
C1C5C10C2×C10C2×C20C2×D20C2×Q8⋊D5 — C4×Q8⋊D5
C5C10C20 — C4×Q8⋊D5
C1C2×C4C42C4×Q8

Generators and relations for C4×Q8⋊D5
 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, ece=b-1c, ede=d-1 >

Subgroups: 454 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], Q8, C23, D5 [×2], C10 [×3], C42, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×2], C20 [×4], D10 [×4], C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C52C8 [×2], C52C8, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C4×SD16, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, Q8⋊D5 [×4], C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C4×C52C8, C20.Q8, D206C4, Q8⋊Dic5, C4×D20, C2×Q8⋊D5, Q8×C20, C4×Q8⋊D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, SD16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×SD16, C4○D8, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C4×SD16, Q8⋊D5 [×2], C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C2×Q8⋊D5, D4.8D10, C4×Q8⋊D5

Smallest permutation representation of C4×Q8⋊D5
On 160 points
Generators in S160
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 141 101 121)(82 142 102 122)(83 143 103 123)(84 144 104 124)(85 145 105 125)(86 146 106 126)(87 147 107 127)(88 148 108 128)(89 149 109 129)(90 150 110 130)(91 151 111 131)(92 152 112 132)(93 153 113 133)(94 154 114 134)(95 155 115 135)(96 156 116 136)(97 157 117 137)(98 158 118 138)(99 159 119 139)(100 160 120 140)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 106 6 101)(2 107 7 102)(3 108 8 103)(4 109 9 104)(5 110 10 105)(11 116 16 111)(12 117 17 112)(13 118 18 113)(14 119 19 114)(15 120 20 115)(21 86 26 81)(22 87 27 82)(23 88 28 83)(24 89 29 84)(25 90 30 85)(31 96 36 91)(32 97 37 92)(33 98 38 93)(34 99 39 94)(35 100 40 95)(41 146 46 141)(42 147 47 142)(43 148 48 143)(44 149 49 144)(45 150 50 145)(51 156 56 151)(52 157 57 152)(53 158 58 153)(54 159 59 154)(55 160 60 155)(61 126 66 121)(62 127 67 122)(63 128 68 123)(64 129 69 124)(65 130 70 125)(71 136 76 131)(72 137 77 132)(73 138 78 133)(74 139 79 134)(75 140 80 135)
(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 5)(2 4)(6 10)(7 9)(11 20)(12 19)(13 18)(14 17)(15 16)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 65)(62 64)(66 70)(67 69)(71 80)(72 79)(73 78)(74 77)(75 76)(81 95)(82 94)(83 93)(84 92)(85 91)(86 100)(87 99)(88 98)(89 97)(90 96)(101 115)(102 114)(103 113)(104 112)(105 111)(106 120)(107 119)(108 118)(109 117)(110 116)(121 135)(122 134)(123 133)(124 132)(125 131)(126 140)(127 139)(128 138)(129 137)(130 136)(141 155)(142 154)(143 153)(144 152)(145 151)(146 160)(147 159)(148 158)(149 157)(150 156)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A···8H10A···10F20A···20H20I···20AF
order12222244444444444444558···810···1020···2020···20
size1111202011112222444420202210···102···22···24···4

68 irreducible representations

dim1111111112222222222244
type++++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D5SD16C4○D4D10D10D10C4○D8C5⋊D4C4×D5C4○D20Q8⋊D5D4.8D10
kernelC4×Q8⋊D5C4×C52C8C20.Q8D206C4Q8⋊Dic5C4×D20C2×Q8⋊D5Q8×C20Q8⋊D5C2×C20C4×Q8C20C20C42C4⋊C4C2×Q8C10C2×C4Q8C4C4C2
# reps1111111182242222488844

Matrix representation of C4×Q8⋊D5 in GL4(𝔽41) generated by

32000
03200
00320
00032
,
40000
04000
00401
00391
,
184000
362300
00026
00110
,
354000
364000
0010
0001
,
0700
6000
00400
00391
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[40,0,0,0,0,40,0,0,0,0,40,39,0,0,1,1],[18,36,0,0,40,23,0,0,0,0,0,11,0,0,26,0],[35,36,0,0,40,40,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,7,0,0,0,0,0,40,39,0,0,0,1] >;

C4×Q8⋊D5 in GAP, Magma, Sage, TeX

C_4\times Q_8\rtimes D_5
% in TeX

G:=Group("C4xQ8:D5");
// GroupNames label

G:=SmallGroup(320,652);
// by ID

G=gap.SmallGroup(320,652);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,58,1684,851,102,12550]);
// 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,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

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