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## G = C22×C5⋊2C8order 160 = 25·5

### Direct product of C22 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C22×C5⋊2C8
 Lower central C5 — C22×C5⋊2C8
 Upper central C1 — C22×C4

Generators and relations for C22×C52C8
G = < a,b,c,d | a2=b2=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 120 in 76 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, C22×C4, C20, C20, C2×C10, C22×C8, C52C8, C2×C20, C22×C10, C2×C52C8, C22×C20, C22×C52C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, Dic5, D10, C22×C8, C52C8, C2×Dic5, C22×D5, C2×C52C8, C22×Dic5, C22×C52C8

Smallest permutation representation of C22×C52C8
Regular action on 160 points
Generators in S160
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)(41 142)(42 143)(43 144)(44 137)(45 138)(46 139)(47 140)(48 141)(49 150)(50 151)(51 152)(52 145)(53 146)(54 147)(55 148)(56 149)(57 158)(58 159)(59 160)(60 153)(61 154)(62 155)(63 156)(64 157)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(81 126)(82 127)(83 128)(84 121)(85 122)(86 123)(87 124)(88 125)(89 134)(90 135)(91 136)(92 129)(93 130)(94 131)(95 132)(96 133)
(1 144)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 80)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 110)(34 111)(35 112)(36 105)(37 106)(38 107)(39 108)(40 109)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 121)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 129)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 145)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 153)(97 120)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)
(1 11 65 147 155)(2 156 148 66 12)(3 13 67 149 157)(4 158 150 68 14)(5 15 69 151 159)(6 160 152 70 16)(7 9 71 145 153)(8 154 146 72 10)(17 86 94 140 78)(18 79 141 95 87)(19 88 96 142 80)(20 73 143 89 81)(21 82 90 144 74)(22 75 137 91 83)(23 84 92 138 76)(24 77 139 93 85)(25 112 117 50 58)(26 59 51 118 105)(27 106 119 52 60)(28 61 53 120 107)(29 108 113 54 62)(30 63 55 114 109)(31 110 115 56 64)(32 57 49 116 111)(33 100 121 129 45)(34 46 130 122 101)(35 102 123 131 47)(36 48 132 124 103)(37 104 125 133 41)(38 42 134 126 97)(39 98 127 135 43)(40 44 136 128 99)
(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)

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

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

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

64 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5A 5B 8A ··· 8P 10A ··· 10N 20A ··· 20P order 1 2 ··· 2 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 2 5 ··· 5 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 D5 Dic5 D10 Dic5 C5⋊2C8 kernel C22×C5⋊2C8 C2×C5⋊2C8 C22×C20 C2×C20 C22×C10 C2×C10 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 16 2 6 6 2 16

Matrix representation of C22×C52C8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 35 40 0 0 36 40
,
 27 0 0 0 0 1 0 0 0 0 33 6 0 0 10 8
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,35,36,0,0,40,40],[27,0,0,0,0,1,0,0,0,0,33,10,0,0,6,8] >;

C22×C52C8 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes_2C_8
% in TeX

G:=Group("C2^2xC5:2C8");
// GroupNames label

G:=SmallGroup(160,141);
// by ID

G=gap.SmallGroup(160,141);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,69,4613]);
// Polycyclic

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

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