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## G = C5×C8.2D4order 320 = 26·5

### Direct product of C5 and C8.2D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C8.2D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — Q8×C10 — C10×SD16 — C5×C8.2D4
 Lower central C1 — C2 — C2×C4 — C5×C8.2D4
 Upper central C1 — C2×C10 — C4×C20 — C5×C8.2D4

Generators and relations for C5×C8.2D4
G = < a,b,c,d | a5=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b3, dcd=c3 >

Subgroups: 226 in 124 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C8.2D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C5×Q16, D4×C10, Q8×C10, Q8×C10, C5×C8⋊C4, C5×C4.4D4, C5×C4⋊Q8, C10×SD16, C10×Q16, C5×C8.2D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C41D4, C8.C22, C5×D4, C22×C10, C8.2D4, D4×C10, C5×C41D4, C5×C8.C22, C5×C8.2D4

Smallest permutation representation of C5×C8.2D4
On 160 points
Generators in S160
(1 90 50 82 42)(2 91 51 83 43)(3 92 52 84 44)(4 93 53 85 45)(5 94 54 86 46)(6 95 55 87 47)(7 96 56 88 48)(8 89 49 81 41)(9 115 139 17 131)(10 116 140 18 132)(11 117 141 19 133)(12 118 142 20 134)(13 119 143 21 135)(14 120 144 22 136)(15 113 137 23 129)(16 114 138 24 130)(25 128 155 33 147)(26 121 156 34 148)(27 122 157 35 149)(28 123 158 36 150)(29 124 159 37 151)(30 125 160 38 152)(31 126 153 39 145)(32 127 154 40 146)(57 75 107 66 101)(58 76 108 67 102)(59 77 109 68 103)(60 78 110 69 104)(61 79 111 70 97)(62 80 112 71 98)(63 73 105 72 99)(64 74 106 65 100)
(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)
(1 108 31 118 5 112 27 114)(2 105 32 115 6 109 28 119)(3 110 25 120 7 106 29 116)(4 107 26 117 8 111 30 113)(9 47 77 150 13 43 73 146)(10 44 78 147 14 48 74 151)(11 41 79 152 15 45 75 148)(12 46 80 149 16 42 76 145)(17 55 103 158 21 51 99 154)(18 52 104 155 22 56 100 159)(19 49 97 160 23 53 101 156)(20 54 98 157 24 50 102 153)(33 136 88 64 37 132 84 60)(34 133 81 61 38 129 85 57)(35 130 82 58 39 134 86 62)(36 135 83 63 40 131 87 59)(65 124 140 92 69 128 144 96)(66 121 141 89 70 125 137 93)(67 126 142 94 71 122 138 90)(68 123 143 91 72 127 139 95)
(1 31)(2 26)(3 29)(4 32)(5 27)(6 30)(7 25)(8 28)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(33 88)(34 83)(35 86)(36 81)(37 84)(38 87)(39 82)(40 85)(41 150)(42 145)(43 148)(44 151)(45 146)(46 149)(47 152)(48 147)(49 158)(50 153)(51 156)(52 159)(53 154)(54 157)(55 160)(56 155)(57 59)(58 62)(61 63)(66 68)(67 71)(70 72)(73 79)(75 77)(76 80)(89 123)(90 126)(91 121)(92 124)(93 127)(94 122)(95 125)(96 128)(97 99)(98 102)(101 103)(105 111)(107 109)(108 112)(113 119)(115 117)(116 120)(129 135)(131 133)(132 136)(137 143)(139 141)(140 144)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M 10N 10O 10P 20A ··· 20H 20I ··· 20P 20Q ··· 20AB 40A ··· 40P order 1 2 2 2 2 4 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 8 2 2 4 4 8 8 8 1 1 1 1 4 4 4 4 1 ··· 1 8 8 8 8 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

80 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 C8.C22 C5×C8.C22 kernel C5×C8.2D4 C5×C8⋊C4 C5×C4.4D4 C5×C4⋊Q8 C10×SD16 C10×Q16 C8.2D4 C8⋊C4 C4.4D4 C4⋊Q8 C2×SD16 C2×Q16 C40 C2×C20 C8 C2×C4 C10 C2 # reps 1 1 1 1 2 2 4 4 4 4 8 8 4 2 16 8 2 8

Matrix representation of C5×C8.2D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 32 2 0 0 0 0 0 9 0 0 0 0 0 0 31 10 28 13 0 0 31 31 28 28 0 0 3 38 10 31 0 0 3 3 10 10
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 33 8 6 35 0 0 33 33 6 6 0 0 20 21 8 33 0 0 20 20 8 8
,
 1 0 0 0 0 0 9 40 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[32,0,0,0,0,0,2,9,0,0,0,0,0,0,31,31,3,3,0,0,10,31,38,3,0,0,28,28,10,10,0,0,13,28,31,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,33,33,20,20,0,0,8,33,21,20,0,0,6,6,8,8,0,0,35,6,33,8],[1,9,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

C5×C8.2D4 in GAP, Magma, Sage, TeX

C_5\times C_8._2D_4
% in TeX

G:=Group("C5xC8.2D4");
// GroupNames label

G:=SmallGroup(320,998);
// by ID

G=gap.SmallGroup(320,998);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,288,1766,1731,436,7004,172]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^3,d*c*d=c^3>;
// generators/relations

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