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## G = C5×C42.C22order 320 = 26·5

### Direct product of C5 and C42.C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C42.C22
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C20 — C5×C8⋊C4 — C5×C42.C22
 Lower central C1 — C22 — C2×C4 — C5×C42.C22
 Upper central C1 — C2×C10 — C4×C20 — C5×C42.C22

Generators and relations for C5×C42.C22
G = < a,b,c,d,e | a5=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc2, ebe=b-1, cd=dc, ece=b2c-1, ede=b-1c2d >

Subgroups: 146 in 70 conjugacy classes, 30 normal (18 characteristic)
C1, C2, C2 [×2], C2, C4 [×4], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4, Q8, C23, C10, C10 [×2], C10, C42, C22⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, C20 [×4], C2×C10, C2×C10 [×3], C8⋊C4 [×2], C4.4D4, C40 [×4], C2×C20, C2×C20 [×2], C2×C20, C5×D4, C5×Q8, C22×C10, C42.C22, C4×C20, C5×C22⋊C4 [×2], C2×C40 [×2], D4×C10, Q8×C10, C5×C8⋊C4 [×2], C5×C4.4D4, C5×C42.C22
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, C20 [×2], C2×C10, C4.D4, C4≀C2 [×2], C2×C20, C5×D4 [×2], C42.C22, C5×C22⋊C4, C5×C4.D4, C5×C4≀C2 [×2], C5×C42.C22

Smallest permutation representation of C5×C42.C22
On 160 points
Generators in S160
(1 47 39 31 23)(2 48 40 32 24)(3 41 33 25 17)(4 42 34 26 18)(5 43 35 27 19)(6 44 36 28 20)(7 45 37 29 21)(8 46 38 30 22)(9 113 105 97 89)(10 114 106 98 90)(11 115 107 99 91)(12 116 108 100 92)(13 117 109 101 93)(14 118 110 102 94)(15 119 111 103 95)(16 120 112 104 96)(49 88 73 65 57)(50 81 74 66 58)(51 82 75 67 59)(52 83 76 68 60)(53 84 77 69 61)(54 85 78 70 62)(55 86 79 71 63)(56 87 80 72 64)(121 153 145 137 129)(122 154 146 138 130)(123 155 147 139 131)(124 156 148 140 132)(125 157 149 141 133)(126 158 150 142 134)(127 159 151 143 135)(128 160 152 144 136)
(1 55 127 95)(2 52 128 92)(3 49 121 89)(4 54 122 94)(5 51 123 91)(6 56 124 96)(7 53 125 93)(8 50 126 90)(9 41 88 153)(10 46 81 158)(11 43 82 155)(12 48 83 160)(13 45 84 157)(14 42 85 154)(15 47 86 159)(16 44 87 156)(17 57 129 97)(18 62 130 102)(19 59 131 99)(20 64 132 104)(21 61 133 101)(22 58 134 98)(23 63 135 103)(24 60 136 100)(25 65 137 105)(26 70 138 110)(27 67 139 107)(28 72 140 112)(29 69 141 109)(30 66 142 106)(31 71 143 111)(32 68 144 108)(33 73 145 113)(34 78 146 118)(35 75 147 115)(36 80 148 120)(37 77 149 117)(38 74 150 114)(39 79 151 119)(40 76 152 116)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(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)
(2 92)(3 125)(4 50)(6 96)(7 121)(8 54)(9 13)(10 154)(11 82)(12 48)(14 158)(15 86)(16 44)(17 133)(18 58)(20 104)(21 129)(22 62)(24 100)(25 141)(26 66)(28 112)(29 137)(30 70)(32 108)(33 149)(34 74)(36 120)(37 145)(38 78)(40 116)(41 157)(42 81)(45 153)(46 85)(49 53)(51 91)(52 128)(55 95)(56 124)(57 61)(59 99)(60 136)(63 103)(64 132)(65 69)(67 107)(68 144)(71 111)(72 140)(73 77)(75 115)(76 152)(79 119)(80 148)(83 160)(84 88)(87 156)(89 93)(90 122)(94 126)(97 101)(98 130)(102 134)(105 109)(106 138)(110 142)(113 117)(114 146)(118 150)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M 10N 10O 10P 20A ··· 20P 20Q 20R 20S 20T 20U 20V 20W 20X 40A ··· 40AF order 1 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 8 2 2 2 2 4 8 1 1 1 1 4 ··· 4 1 ··· 1 8 8 8 8 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 D4 C4≀C2 C5×D4 C5×C4≀C2 C4.D4 C5×C4.D4 kernel C5×C42.C22 C5×C8⋊C4 C5×C4.4D4 D4×C10 Q8×C10 C42.C22 C8⋊C4 C4.4D4 C2×D4 C2×Q8 C2×C20 C10 C2×C4 C2 C10 C2 # reps 1 2 1 2 2 4 8 4 8 8 2 8 8 32 1 4

Matrix representation of C5×C42.C22 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 37 0 0 0 0 37
,
 0 1 0 0 40 0 0 0 0 0 1 9 0 0 18 40
,
 9 0 0 0 0 9 0 0 0 0 32 1 0 0 2 9
,
 36 5 0 0 5 5 0 0 0 0 8 36 0 0 31 0
,
 1 0 0 0 0 40 0 0 0 0 1 9 0 0 0 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[0,40,0,0,1,0,0,0,0,0,1,18,0,0,9,40],[9,0,0,0,0,9,0,0,0,0,32,2,0,0,1,9],[36,5,0,0,5,5,0,0,0,0,8,31,0,0,36,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,9,40] >;

C5×C42.C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2.C_2^2
% in TeX

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

G:=SmallGroup(320,134);
// by ID

G=gap.SmallGroup(320,134);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2530,248,4911,102]);
// Polycyclic

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

׿
×
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