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## G = C5×Q8⋊C4order 160 = 25·5

### Direct product of C5 and Q8⋊C4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×Q8⋊C4
G = < a,b,c,d | a5=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Smallest permutation representation of C5×Q8⋊C4
Regular action 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 54 29 12)(2 55 30 13)(3 51 26 14)(4 52 27 15)(5 53 28 11)(6 22 136 143)(7 23 137 144)(8 24 138 145)(9 25 139 141)(10 21 140 142)(16 32 157 153)(17 33 158 154)(18 34 159 155)(19 35 160 151)(20 31 156 152)(36 41 70 46)(37 42 66 47)(38 43 67 48)(39 44 68 49)(40 45 69 50)(56 93 85 63)(57 94 81 64)(58 95 82 65)(59 91 83 61)(60 92 84 62)(71 80 90 109)(72 76 86 110)(73 77 87 106)(74 78 88 107)(75 79 89 108)(96 133 125 103)(97 134 121 104)(98 135 122 105)(99 131 123 101)(100 132 124 102)(111 120 130 149)(112 116 126 150)(113 117 127 146)(114 118 128 147)(115 119 129 148)
(1 77 29 106)(2 78 30 107)(3 79 26 108)(4 80 27 109)(5 76 28 110)(6 126 136 112)(7 127 137 113)(8 128 138 114)(9 129 139 115)(10 130 140 111)(11 86 53 72)(12 87 54 73)(13 88 55 74)(14 89 51 75)(15 90 52 71)(16 134 157 104)(17 135 158 105)(18 131 159 101)(19 132 160 102)(20 133 156 103)(21 120 142 149)(22 116 143 150)(23 117 144 146)(24 118 145 147)(25 119 141 148)(31 96 152 125)(32 97 153 121)(33 98 154 122)(34 99 155 123)(35 100 151 124)(36 85 70 56)(37 81 66 57)(38 82 67 58)(39 83 68 59)(40 84 69 60)(41 93 46 63)(42 94 47 64)(43 95 48 65)(44 91 49 61)(45 92 50 62)
(1 146 66 121)(2 147 67 122)(3 148 68 123)(4 149 69 124)(5 150 70 125)(6 85 20 110)(7 81 16 106)(8 82 17 107)(9 83 18 108)(10 84 19 109)(11 112 41 103)(12 113 42 104)(13 114 43 105)(14 115 44 101)(15 111 45 102)(21 92 35 90)(22 93 31 86)(23 94 32 87)(24 95 33 88)(25 91 34 89)(26 119 39 99)(27 120 40 100)(28 116 36 96)(29 117 37 97)(30 118 38 98)(46 133 53 126)(47 134 54 127)(48 135 55 128)(49 131 51 129)(50 132 52 130)(56 156 76 136)(57 157 77 137)(58 158 78 138)(59 159 79 139)(60 160 80 140)(61 155 75 141)(62 151 71 142)(63 152 72 143)(64 153 73 144)(65 154 74 145)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D4 D4 SD16 Q16 C5×D4 C5×D4 C5×SD16 C5×Q16 kernel C5×Q8⋊C4 C5×C4⋊C4 C2×C40 Q8×C10 C5×Q8 Q8⋊C4 C4⋊C4 C2×C8 C2×Q8 Q8 C20 C2×C10 C10 C10 C4 C22 C2 C2 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2 2 4 4 8 8

Matrix representation of C5×Q8⋊C4 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 37 0 0 0 0 37
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 0 40 0 0 40 0 0 0 0 0 40 11 0 0 11 1
,
 30 32 0 0 9 11 0 0 0 0 38 21 0 0 21 3
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,40,11,0,0,11,1],[30,9,0,0,32,11,0,0,0,0,38,21,0,0,21,3] >;

C5×Q8⋊C4 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes C_4
% in TeX

G:=Group("C5xQ8:C4");
// GroupNames label

G:=SmallGroup(160,53);
// by ID

G=gap.SmallGroup(160,53);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,487,2403,1209,117]);
// Polycyclic

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

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