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## G = C5×C2.C42order 160 = 25·5

### Direct product of C5 and C2.C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C5×C2.C42
 Lower central C1 — C2 — C5×C2.C42
 Upper central C1 — C22×C10 — C5×C2.C42

Generators and relations for C5×C2.C42
G = < a,b,c,d | a5=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 100 in 76 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C22×C4, C20, C2×C10, C2×C10, C2.C42, C2×C20, C2×C20, C22×C10, C22×C20, C5×C2.C42
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2.C42, C2×C20, C5×D4, C5×Q8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C2.C42

Smallest permutation representation of C5×C2.C42
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 45)(2 41)(3 42)(4 43)(5 44)(6 140)(7 136)(8 137)(9 138)(10 139)(11 53)(12 54)(13 55)(14 51)(15 52)(16 141)(17 142)(18 143)(19 144)(20 145)(21 155)(22 151)(23 152)(24 153)(25 154)(26 69)(27 70)(28 66)(29 67)(30 68)(31 156)(32 157)(33 158)(34 159)(35 160)(36 46)(37 47)(38 48)(39 49)(40 50)(56 81)(57 82)(58 83)(59 84)(60 85)(61 95)(62 91)(63 92)(64 93)(65 94)(71 109)(72 110)(73 106)(74 107)(75 108)(76 86)(77 87)(78 88)(79 89)(80 90)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 146 66 121)(2 147 67 122)(3 148 68 123)(4 149 69 124)(5 150 70 125)(6 85 23 110)(7 81 24 106)(8 82 25 107)(9 83 21 108)(10 84 22 109)(11 116 36 104)(12 117 37 105)(13 118 38 101)(14 119 39 102)(15 120 40 103)(16 95 33 88)(17 91 34 89)(18 92 35 90)(19 93 31 86)(20 94 32 87)(26 99 43 111)(27 100 44 112)(28 96 45 113)(29 97 41 114)(30 98 42 115)(46 133 53 126)(47 134 54 127)(48 135 55 128)(49 131 51 129)(50 132 52 130)(56 153 73 136)(57 154 74 137)(58 155 75 138)(59 151 71 139)(60 152 72 140)(61 158 78 141)(62 159 79 142)(63 160 80 143)(64 156 76 144)(65 157 77 145)
(1 65 12 81)(2 61 13 82)(3 62 14 83)(4 63 15 84)(5 64 11 85)(6 100 144 133)(7 96 145 134)(8 97 141 135)(9 98 142 131)(10 99 143 132)(16 101 137 122)(17 102 138 123)(18 103 139 124)(19 104 140 125)(20 105 136 121)(21 115 159 129)(22 111 160 130)(23 112 156 126)(24 113 157 127)(25 114 158 128)(26 90 50 71)(27 86 46 72)(28 87 47 73)(29 88 48 74)(30 89 49 75)(31 116 152 150)(32 117 153 146)(33 118 154 147)(34 119 155 148)(35 120 151 149)(36 110 70 76)(37 106 66 77)(38 107 67 78)(39 108 68 79)(40 109 69 80)(41 95 55 57)(42 91 51 58)(43 92 52 59)(44 93 53 60)(45 94 54 56)

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

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

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

100 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 5A 5B 5C 5D 10A ··· 10AB 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C5 C10 C20 D4 Q8 C5×D4 C5×Q8 kernel C5×C2.C42 C22×C20 C2×C20 C2.C42 C22×C4 C2×C4 C2×C10 C2×C10 C22 C22 # reps 1 3 12 4 12 48 3 1 12 4

Matrix representation of C5×C2.C42 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 9 0 0 0 0 9 0 0 0 0 3 25 0 0 16 38
,
 9 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,3,16,0,0,25,38],[9,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C5×C2.C42 in GAP, Magma, Sage, TeX

C_5\times C_2.C_4^2
% in TeX

G:=Group("C5xC2.C4^2");
// GroupNames label

G:=SmallGroup(160,45);
// by ID

G=gap.SmallGroup(160,45);
# by ID

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

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

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