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## G = C2×C5⋊Q16order 160 = 25·5

### Direct product of C2 and C5⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C5⋊Q16
 Chief series C1 — C5 — C10 — C20 — Dic10 — C2×Dic10 — C2×C5⋊Q16
 Lower central C5 — C10 — C20 — C2×C5⋊Q16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×C5⋊Q16
G = < a,b,c,d | a2=b5=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 152 in 60 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C10, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×Q16, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C52C8, C5⋊Q16, C2×Dic10, Q8×C10, C2×C5⋊Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C2×Q16, C5⋊D4, C22×D5, C5⋊Q16, C2×C5⋊D4, C2×C5⋊Q16

Smallest permutation representation of C2×C5⋊Q16
Regular action on 160 points
Generators in S160
(1 78)(2 79)(3 80)(4 73)(5 74)(6 75)(7 76)(8 77)(9 151)(10 152)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(41 120)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 100)(50 101)(51 102)(52 103)(53 104)(54 97)(55 98)(56 99)(57 108)(58 109)(59 110)(60 111)(61 112)(62 105)(63 106)(64 107)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 153)(144 154)
(1 49 39 146 63)(2 64 147 40 50)(3 51 33 148 57)(4 58 149 34 52)(5 53 35 150 59)(6 60 151 36 54)(7 55 37 152 61)(8 62 145 38 56)(9 21 97 75 111)(10 112 76 98 22)(11 23 99 77 105)(12 106 78 100 24)(13 17 101 79 107)(14 108 80 102 18)(15 19 103 73 109)(16 110 74 104 20)(25 143 86 94 48)(26 41 95 87 144)(27 137 88 96 42)(28 43 89 81 138)(29 139 82 90 44)(30 45 91 83 140)(31 141 84 92 46)(32 47 93 85 142)(65 153 126 134 119)(66 120 135 127 154)(67 155 128 136 113)(68 114 129 121 156)(69 157 122 130 115)(70 116 131 123 158)(71 159 124 132 117)(72 118 133 125 160)
(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)
(1 115 5 119)(2 114 6 118)(3 113 7 117)(4 120 8 116)(9 85 13 81)(10 84 14 88)(11 83 15 87)(12 82 16 86)(17 138 21 142)(18 137 22 141)(19 144 23 140)(20 143 24 139)(25 100 29 104)(26 99 30 103)(27 98 31 102)(28 97 32 101)(33 155 37 159)(34 154 38 158)(35 153 39 157)(36 160 40 156)(41 77 45 73)(42 76 46 80)(43 75 47 79)(44 74 48 78)(49 69 53 65)(50 68 54 72)(51 67 55 71)(52 66 56 70)(57 136 61 132)(58 135 62 131)(59 134 63 130)(60 133 64 129)(89 111 93 107)(90 110 94 106)(91 109 95 105)(92 108 96 112)(121 151 125 147)(122 150 126 146)(123 149 127 145)(124 148 128 152)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 20 20 2 2 10 10 10 10 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 D4 D4 D5 Q16 D10 D10 C5⋊D4 C5⋊D4 C5⋊Q16 kernel C2×C5⋊Q16 C2×C5⋊2C8 C5⋊Q16 C2×Dic10 Q8×C10 C20 C2×C10 C2×Q8 C10 C2×C4 Q8 C4 C22 C2 # reps 1 1 4 1 1 1 1 2 4 2 4 4 4 4

Matrix representation of C2×C5⋊Q16 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 1 0
,
 1 0 0 0 0 0 29 12 0 0 0 29 29 0 0 0 0 0 1 7 0 0 0 0 40
,
 1 0 0 0 0 0 32 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,1,0,0,0,40,0],[1,0,0,0,0,0,29,29,0,0,0,12,29,0,0,0,0,0,1,0,0,0,0,7,40],[1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2×C5⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes Q_{16}
% in TeX

G:=Group("C2xC5:Q16");
// GroupNames label

G:=SmallGroup(160,164);
// by ID

G=gap.SmallGroup(160,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,86,579,159,69,4613]);
// Polycyclic

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

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