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G = C5×D46D4order 320 = 26·5

Direct product of C5 and D46D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×D46D4, C10.1172- 1+4, D46(C5×D4), (C5×D4)⋊24D4, C4⋊Q813C10, (C4×D4)⋊16C10, (D4×C20)⋊45C2, C4.41(D4×C10), C2016(C4○D4), C4⋊D412C10, C20.402(C2×D4), C22⋊Q812C10, C22.6(D4×C10), C42.42(C2×C10), (C2×C20).675C23, (C4×C20).283C22, (C2×C10).367C24, C22.D49C10, C10.195(C22×D4), C2.9(C5×2- 1+4), (D4×C10).321C22, (C22×C10).99C23, C23.41(C22×C10), C22.41(C23×C10), (Q8×C10).273C22, (C22×C20).453C22, C42(C5×C4○D4), (C5×C4⋊Q8)⋊34C2, (C10×C4⋊C4)⋊47C2, (C2×C4⋊C4)⋊20C10, C2.19(D4×C2×C10), (C2×C4○D4)⋊7C10, (C10×C4○D4)⋊23C2, C4⋊C4.31(C2×C10), (C5×C4⋊D4)⋊39C2, C2.21(C10×C4○D4), (C5×C22⋊Q8)⋊39C2, (C2×D4).66(C2×C10), C10.240(C2×C4○D4), (C2×C10).183(C2×D4), C22⋊C4.5(C2×C10), (C2×Q8).60(C2×C10), (C5×C4⋊C4).395C22, (C2×C4).33(C22×C10), (C22×C4).65(C2×C10), (C5×C22.D4)⋊28C2, (C5×C22⋊C4).87C22, SmallGroup(320,1549)

Series: Derived Chief Lower central Upper central

C1C22 — C5×D46D4
C1C2C22C2×C10C2×C20D4×C10C5×C22.D4 — C5×D46D4
C1C22 — C5×D46D4
C1C2×C10 — C5×D46D4

Generators and relations for C5×D46D4
 G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 426 in 292 conjugacy classes, 166 normal (34 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C5, C2×C4 [×3], C2×C4 [×8], C2×C4 [×16], D4 [×4], D4 [×10], Q8 [×4], C23 [×4], C10 [×3], C10 [×6], C42, C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C20 [×4], C20 [×9], C2×C10, C2×C10 [×4], C2×C10 [×10], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], C2×C20 [×3], C2×C20 [×8], C2×C20 [×16], C5×D4 [×4], C5×D4 [×10], C5×Q8 [×4], C22×C10 [×4], D46D4, C4×C20, C5×C22⋊C4 [×8], C5×C4⋊C4 [×2], C5×C4⋊C4 [×8], C22×C20 [×8], D4×C10 [×2], D4×C10 [×4], Q8×C10 [×2], C5×C4○D4 [×8], C10×C4⋊C4 [×2], D4×C20 [×2], C5×C4⋊D4 [×2], C5×C22⋊Q8 [×2], C5×C22.D4 [×4], C5×C4⋊Q8, C10×C4○D4 [×2], C5×D46D4
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22×D4, C2×C4○D4, 2- 1+4, C5×D4 [×4], C22×C10 [×15], D46D4, D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C2×C10, C10×C4○D4, C5×2- 1+4, C5×D46D4

Smallest permutation representation of C5×D46D4
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)
(1 135 35 116)(2 131 31 117)(3 132 32 118)(4 133 33 119)(5 134 34 120)(6 74 160 60)(7 75 156 56)(8 71 157 57)(9 72 158 58)(10 73 159 59)(11 62 17 68)(12 63 18 69)(13 64 19 70)(14 65 20 66)(15 61 16 67)(21 127 27 121)(22 128 28 122)(23 129 29 123)(24 130 30 124)(25 126 26 125)(36 115 55 96)(37 111 51 97)(38 112 52 98)(39 113 53 99)(40 114 54 100)(41 107 47 101)(42 108 48 102)(43 109 49 103)(44 110 50 104)(45 106 46 105)(76 155 95 136)(77 151 91 137)(78 152 92 138)(79 153 93 139)(80 154 94 140)(81 147 87 141)(82 148 88 142)(83 149 89 143)(84 150 90 144)(85 146 86 145)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 86)(7 87)(8 88)(9 89)(10 90)(11 77)(12 78)(13 79)(14 80)(15 76)(16 95)(17 91)(18 92)(19 93)(20 94)(21 96)(22 97)(23 98)(24 99)(25 100)(26 114)(27 115)(28 111)(29 112)(30 113)(31 108)(32 109)(33 110)(34 106)(35 107)(36 121)(37 122)(38 123)(39 124)(40 125)(41 116)(42 117)(43 118)(44 119)(45 120)(46 134)(47 135)(48 131)(49 132)(50 133)(51 128)(52 129)(53 130)(54 126)(55 127)(56 141)(57 142)(58 143)(59 144)(60 145)(61 136)(62 137)(63 138)(64 139)(65 140)(66 154)(67 155)(68 151)(69 152)(70 153)(71 148)(72 149)(73 150)(74 146)(75 147)(81 156)(82 157)(83 158)(84 159)(85 160)
(1 15 21 156)(2 11 22 157)(3 12 23 158)(4 13 24 159)(5 14 25 160)(6 34 20 26)(7 35 16 27)(8 31 17 28)(9 32 18 29)(10 33 19 30)(36 147 41 155)(37 148 42 151)(38 149 43 152)(39 150 44 153)(40 146 45 154)(46 140 54 145)(47 136 55 141)(48 137 51 142)(49 138 52 143)(50 139 53 144)(56 135 61 127)(57 131 62 128)(58 132 63 129)(59 133 64 130)(60 134 65 126)(66 125 74 120)(67 121 75 116)(68 122 71 117)(69 123 72 118)(70 124 73 119)(76 96 81 101)(77 97 82 102)(78 98 83 103)(79 99 84 104)(80 100 85 105)(86 106 94 114)(87 107 95 115)(88 108 91 111)(89 109 92 112)(90 110 93 113)
(1 21)(2 22)(3 23)(4 24)(5 25)(26 34)(27 35)(28 31)(29 32)(30 33)(36 47)(37 48)(38 49)(39 50)(40 46)(41 55)(42 51)(43 52)(44 53)(45 54)(76 95)(77 91)(78 92)(79 93)(80 94)(81 87)(82 88)(83 89)(84 90)(85 86)(96 107)(97 108)(98 109)(99 110)(100 106)(101 115)(102 111)(103 112)(104 113)(105 114)(116 121)(117 122)(118 123)(119 124)(120 125)(126 134)(127 135)(128 131)(129 132)(130 133)(136 155)(137 151)(138 152)(139 153)(140 154)(141 147)(142 148)(143 149)(144 150)(145 146)

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

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

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

125 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4O5A5B5C5D10A···10L10M···10AB10AC···10AJ20A···20AF20AG···20BH
order12222222224···44···4555510···1010···1010···1020···2020···20
size11112222442···24···411111···12···24···42···24···4

125 irreducible representations

dim1111111111111111222244
type+++++++++-
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10D4C4○D4C5×D4C5×C4○D42- 1+4C5×2- 1+4
kernelC5×D46D4C10×C4⋊C4D4×C20C5×C4⋊D4C5×C22⋊Q8C5×C22.D4C5×C4⋊Q8C10×C4○D4D46D4C2×C4⋊C4C4×D4C4⋊D4C22⋊Q8C22.D4C4⋊Q8C2×C4○D4C5×D4C20D4C4C10C2
# reps1222241248888164844161614

Matrix representation of C5×D46D4 in GL5(𝔽41)

370000
01000
00100
00010
00001
,
10000
09000
003200
000400
000040
,
400000
00900
032000
00010
00001
,
400000
040000
004000
00062
000235
,
400000
040000
00100
000400
00061

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,32,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,6,2,0,0,0,2,35],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,6,0,0,0,0,1] >;

C5×D46D4 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_6D_4
% in TeX

G:=Group("C5xD4:6D4");
// GroupNames label

G:=SmallGroup(320,1549);
// by ID

G=gap.SmallGroup(320,1549);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,3446,436,1242]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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