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## G = C5×Q8○D8order 320 = 26·5

### Direct product of C5 and Q8○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×Q8○D8
 Chief series C1 — C2 — C4 — C20 — C5×Q8 — C5×Q16 — C10×Q16 — C5×Q8○D8
 Lower central C1 — C2 — C4 — C5×Q8○D8
 Upper central C1 — C10 — C5×C4○D4 — C5×Q8○D8

Generators and relations for C5×Q8○D8
G = < a,b,c,d,e | a5=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 346 in 248 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, Q8○D8, C2×C40, C5×M4(2), C5×D8, C5×SD16, C5×Q16, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4○D4, C5×C8○D4, C10×Q16, C5×C4○D8, C5×C8.C22, C5×2- 1+4, C5×Q8○D8
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C22×D4, C5×D4, C22×C10, Q8○D8, D4×C10, C23×C10, D4×C2×C10, C5×Q8○D8

Smallest permutation representation of C5×Q8○D8
On 160 points
Generators in S160
(1 142 24 153 65)(2 143 17 154 66)(3 144 18 155 67)(4 137 19 156 68)(5 138 20 157 69)(6 139 21 158 70)(7 140 22 159 71)(8 141 23 160 72)(9 81 89 152 75)(10 82 90 145 76)(11 83 91 146 77)(12 84 92 147 78)(13 85 93 148 79)(14 86 94 149 80)(15 87 95 150 73)(16 88 96 151 74)(25 112 119 49 57)(26 105 120 50 58)(27 106 113 51 59)(28 107 114 52 60)(29 108 115 53 61)(30 109 116 54 62)(31 110 117 55 63)(32 111 118 56 64)(33 45 127 135 100)(34 46 128 136 101)(35 47 121 129 102)(36 48 122 130 103)(37 41 123 131 104)(38 42 124 132 97)(39 43 125 133 98)(40 44 126 134 99)
(1 88 5 84)(2 81 6 85)(3 82 7 86)(4 83 8 87)(9 70 13 66)(10 71 14 67)(11 72 15 68)(12 65 16 69)(17 152 21 148)(18 145 22 149)(19 146 23 150)(20 147 24 151)(25 102 29 98)(26 103 30 99)(27 104 31 100)(28 97 32 101)(33 106 37 110)(34 107 38 111)(35 108 39 112)(36 109 40 105)(41 117 45 113)(42 118 46 114)(43 119 47 115)(44 120 48 116)(49 121 53 125)(50 122 54 126)(51 123 55 127)(52 124 56 128)(57 129 61 133)(58 130 62 134)(59 131 63 135)(60 132 64 136)(73 156 77 160)(74 157 78 153)(75 158 79 154)(76 159 80 155)(89 139 93 143)(90 140 94 144)(91 141 95 137)(92 142 96 138)
(1 51 5 55)(2 52 6 56)(3 53 7 49)(4 54 8 50)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 27)(33 78 37 74)(34 79 38 75)(35 80 39 76)(36 73 40 77)(57 144 61 140)(58 137 62 141)(59 138 63 142)(60 139 64 143)(65 113 69 117)(66 114 70 118)(67 115 71 119)(68 116 72 120)(81 128 85 124)(82 121 86 125)(83 122 87 126)(84 123 88 127)(89 136 93 132)(90 129 94 133)(91 130 95 134)(92 131 96 135)(97 152 101 148)(98 145 102 149)(99 146 103 150)(100 147 104 151)(105 156 109 160)(106 157 110 153)(107 158 111 154)(108 159 112 155)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 23)(18 22)(19 21)(25 29)(26 28)(30 32)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)(49 53)(50 52)(54 56)(57 61)(58 60)(62 64)(66 72)(67 71)(68 70)(73 75)(76 80)(77 79)(81 87)(82 86)(83 85)(89 95)(90 94)(91 93)(97 103)(98 102)(99 101)(105 107)(108 112)(109 111)(114 120)(115 119)(116 118)(121 125)(122 124)(126 128)(129 133)(130 132)(134 136)(137 139)(140 144)(141 143)(145 149)(146 148)(150 152)(154 160)(155 159)(156 158)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4J 5A 5B 5C 5D 8A 8B 8C 8D 8E 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10X 20A ··· 20P 20Q ··· 20AN 40A ··· 40H 40I ··· 40T order 1 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 5 5 8 8 8 8 8 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 2 2 2 4 4 2 2 2 2 4 ··· 4 1 1 1 1 2 2 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 C5×D4 C5×D4 Q8○D8 C5×Q8○D8 kernel C5×Q8○D8 C5×C8○D4 C10×Q16 C5×C4○D8 C5×C8.C22 C5×2- 1+4 Q8○D8 C8○D4 C2×Q16 C4○D8 C8.C22 2- 1+4 C5×D4 C5×Q8 D4 Q8 C5 C1 # reps 1 1 3 3 6 2 4 4 12 12 24 8 3 1 12 4 2 8

Matrix representation of C5×Q8○D8 in GL6(𝔽41)

 10 0 0 0 0 0 0 10 0 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 34 40 39 0 0 0 0 40 0 0 0 0 1 0 0 0 0 25 24 7 7
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 23 16 14 28 0 0 0 7 14 0 0 0 0 14 34 0 0 0 25 11 25 18
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 29 29 0 0 0 0 12 29 0 0 0 0 2 2 0 24 0 0 39 0 29 17
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 7 1 1

G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,0,0,25,0,0,34,0,1,24,0,0,40,40,0,7,0,0,39,0,0,7],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,23,0,0,25,0,0,16,7,14,11,0,0,14,14,34,25,0,0,28,0,0,18],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,29,12,2,39,0,0,29,29,2,0,0,0,0,0,0,29,0,0,0,0,24,17],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,7,0,0,0,0,40,1,0,0,0,0,0,1] >;

C5×Q8○D8 in GAP, Magma, Sage, TeX

C_5\times Q_8\circ D_8
% in TeX

G:=Group("C5xQ8oD8");
// GroupNames label

G:=SmallGroup(320,1580);
// by ID

G=gap.SmallGroup(320,1580);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,1128,1193,10085,5052,124]);
// Polycyclic

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

׿
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