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## G = C5×C4.Q8order 160 = 25·5

### Direct product of C5 and C4.Q8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C4.Q8
G = < a,b,c,d | a5=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C5×C4.Q8
Regular action on 160 points
Generators in S160
(1 54 23 11 63)(2 55 24 12 64)(3 56 17 13 57)(4 49 18 14 58)(5 50 19 15 59)(6 51 20 16 60)(7 52 21 9 61)(8 53 22 10 62)(25 85 73 33 65)(26 86 74 34 66)(27 87 75 35 67)(28 88 76 36 68)(29 81 77 37 69)(30 82 78 38 70)(31 83 79 39 71)(32 84 80 40 72)(41 155 119 147 111)(42 156 120 148 112)(43 157 113 149 105)(44 158 114 150 106)(45 159 115 151 107)(46 160 116 152 108)(47 153 117 145 109)(48 154 118 146 110)(89 122 141 97 133)(90 123 142 98 134)(91 124 143 99 135)(92 125 144 100 136)(93 126 137 101 129)(94 127 138 102 130)(95 128 139 103 131)(96 121 140 104 132)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 39 13 35)(10 40 14 36)(11 33 15 37)(12 34 16 38)(17 75 21 79)(18 76 22 80)(19 77 23 73)(20 78 24 74)(41 126 45 122)(42 127 46 123)(43 128 47 124)(44 121 48 125)(49 88 53 84)(50 81 54 85)(51 82 55 86)(52 83 56 87)(57 67 61 71)(58 68 62 72)(59 69 63 65)(60 70 64 66)(89 111 93 107)(90 112 94 108)(91 105 95 109)(92 106 96 110)(97 119 101 115)(98 120 102 116)(99 113 103 117)(100 114 104 118)(129 151 133 147)(130 152 134 148)(131 145 135 149)(132 146 136 150)(137 159 141 155)(138 160 142 156)(139 153 143 157)(140 154 144 158)
(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)
(1 105 31 93)(2 108 32 96)(3 111 25 91)(4 106 26 94)(5 109 27 89)(6 112 28 92)(7 107 29 95)(8 110 30 90)(9 115 37 103)(10 118 38 98)(11 113 39 101)(12 116 40 104)(13 119 33 99)(14 114 34 102)(15 117 35 97)(16 120 36 100)(17 155 73 143)(18 158 74 138)(19 153 75 141)(20 156 76 144)(21 159 77 139)(22 154 78 142)(23 157 79 137)(24 160 80 140)(41 85 124 56)(42 88 125 51)(43 83 126 54)(44 86 127 49)(45 81 128 52)(46 84 121 55)(47 87 122 50)(48 82 123 53)(57 147 65 135)(58 150 66 130)(59 145 67 133)(60 148 68 136)(61 151 69 131)(62 146 70 134)(63 149 71 129)(64 152 72 132)

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

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

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

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

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

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 0
,
 1 1 0 0 39 40 0 0 0 0 26 15 0 0 26 26
,
 32 0 0 0 18 9 0 0 0 0 30 40 0 0 40 11
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[1,39,0,0,1,40,0,0,0,0,26,26,0,0,15,26],[32,18,0,0,0,9,0,0,0,0,30,40,0,0,40,11] >;

C5×C4.Q8 in GAP, Magma, Sage, TeX

C_5\times C_4.Q_8
% in TeX

G:=Group("C5xC4.Q8");
// GroupNames label

G:=SmallGroup(160,56);
// by ID

G=gap.SmallGroup(160,56);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,127,2403,117]);
// Polycyclic

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

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