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## G = C5×C42.C2order 160 = 25·5

### Direct product of C5 and C42.C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C42.C2
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4 — C5×C42.C2
 Lower central C1 — C22 — C5×C42.C2
 Upper central C1 — C2×C10 — C5×C42.C2

Generators and relations for C5×C42.C2
G = < a,b,c,d | a5=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 68 in 56 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, C20, C20, C2×C10, C42.C2, C2×C20, C2×C20, C4×C20, C5×C4⋊C4, C5×C42.C2
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C4○D4, C2×C10, C42.C2, C5×Q8, C22×C10, Q8×C10, C5×C4○D4, C5×C42.C2

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

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 5A 5B 5C 5D 10A ··· 10L 20A ··· 20X 20Y ··· 20AN order 1 2 2 2 4 ··· 4 4 4 4 4 5 5 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C2 C5 C10 C10 Q8 C4○D4 C5×Q8 C5×C4○D4 kernel C5×C42.C2 C4×C20 C5×C4⋊C4 C42.C2 C42 C4⋊C4 C20 C10 C4 C2 # reps 1 1 6 4 4 24 2 4 8 16

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

 16 0 0 0 0 16 0 0 0 0 10 0 0 0 0 10
,
 0 1 0 0 1 0 0 0 0 0 9 0 0 0 0 9
,
 32 0 0 0 0 32 0 0 0 0 32 39 0 0 40 9
,
 30 9 0 0 32 11 0 0 0 0 25 28 0 0 7 16
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[0,1,0,0,1,0,0,0,0,0,9,0,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,32,40,0,0,39,9],[30,32,0,0,9,11,0,0,0,0,25,7,0,0,28,16] >;

C5×C42.C2 in GAP, Magma, Sage, TeX

C_5\times C_4^2.C_2
% in TeX

G:=Group("C5xC4^2.C2");
// GroupNames label

G:=SmallGroup(160,186);
// by ID

G=gap.SmallGroup(160,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,505,487,1514,194]);
// Polycyclic

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

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