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## G = D5×C4.Q8order 320 = 26·5

### Direct product of D5 and C4.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D5×C4.Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — D5×C2×C8 — D5×C4.Q8
 Lower central C5 — C10 — C20 — D5×C4.Q8
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for D5×C4.Q8
G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=c-1d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 478 in 130 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×4], C10, C10 [×2], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8 [×5], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4.Q8, C4.Q8 [×3], C2×C4⋊C4 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C4.Q8, C8×D5 [×4], C2×C52C8, C10.D4 [×2], C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5 [×2], C20.Q8 [×2], C406C4, C5×C4.Q8, D5×C4⋊C4 [×2], D5×C2×C8, D5×C4.Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C4.Q8 [×4], C2×C4⋊C4, C2×SD16 [×2], C4×D5 [×2], C22×D5, C2×C4.Q8, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D5×SD16 [×2], D5×C4.Q8

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 5 5 5 5 2 2 4 4 4 4 10 10 20 20 20 20 2 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + + + - + image C1 C2 C2 C2 C2 C2 C4 Q8 D4 D4 D5 SD16 D10 D10 C4×D5 Q8×D5 D4×D5 D5×SD16 kernel D5×C4.Q8 C20.Q8 C40⋊6C4 C5×C4.Q8 D5×C4⋊C4 D5×C2×C8 C8×D5 C4×D5 C2×Dic5 C22×D5 C4.Q8 D10 C4⋊C4 C2×C8 C8 C4 C22 C2 # reps 1 2 1 1 2 1 8 2 1 1 2 8 4 2 8 2 2 8

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

 7 1 0 0 33 40 0 0 0 0 1 0 0 0 0 1
,
 40 40 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 1 37 0 0 21 40
,
 40 0 0 0 0 40 0 0 0 0 0 19 0 0 13 30
,
 32 0 0 0 0 32 0 0 0 0 33 27 0 0 25 8
G:=sub<GL(4,GF(41))| [7,33,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,40,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,21,0,0,37,40],[40,0,0,0,0,40,0,0,0,0,0,13,0,0,19,30],[32,0,0,0,0,32,0,0,0,0,33,25,0,0,27,8] >;

D5×C4.Q8 in GAP, Magma, Sage, TeX

D_5\times C_4.Q_8
% in TeX

G:=Group("D5xC4.Q8");
// GroupNames label

G:=SmallGroup(320,486);
// by ID

G=gap.SmallGroup(320,486);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,555,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^4=c^2,e^2=c^-1*d^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations

׿
×
𝔽