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## G = C5×C42.6C22order 320 = 26·5

### Direct product of C5 and C42.6C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C42.6C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C2×C40 — C5×C4⋊C8 — C5×C42.6C22
 Lower central C1 — C22 — C5×C42.6C22
 Upper central C1 — C2×C20 — C5×C42.6C22

Generators and relations for C5×C42.6C22
G = < a,b,c,d,e | a5=b4=c4=1, d2=c, e2=b2c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe-1=bc2, cd=dc, ce=ec, ede-1=b2c2d >

Subgroups: 146 in 114 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C42.6C22, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×M4(2), C22×C20, C5×C4⋊C8, C5×C42⋊C2, C22×C40, C10×M4(2), C5×C42.6C22
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C4⋊C4, C8○D4, C2×C20, C5×D4, C5×Q8, C22×C10, C42.6C22, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×C8○D4, C5×C42.6C22

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

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 5C 5D 8A ··· 8H 8I 8J 8K 8L 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20X 20Y ··· 20AN 40A ··· 40AF 40AG ··· 40AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C5 C10 C10 C10 C10 C20 C20 D4 Q8 C8○D4 C5×D4 C5×Q8 C5×C8○D4 kernel C5×C42.6C22 C5×C4⋊C8 C5×C42⋊C2 C22×C40 C10×M4(2) C5×C22⋊C4 C5×C4⋊C4 C42.6C22 C4⋊C8 C42⋊C2 C22×C8 C2×M4(2) C22⋊C4 C4⋊C4 C2×C20 C2×C20 C10 C2×C4 C2×C4 C2 # reps 1 4 1 1 1 4 4 4 16 4 4 4 16 16 2 2 8 8 8 32

Matrix representation of C5×C42.6C22 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 10 0 0 0 0 10
,
 9 0 0 0 0 32 0 0 0 0 7 39 0 0 25 34
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 0 1 0 0 1 0 0 0 0 0 3 0 0 0 0 3
,
 32 0 0 0 0 9 0 0 0 0 7 39 0 0 24 34
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[9,0,0,0,0,32,0,0,0,0,7,25,0,0,39,34],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,3,0,0,0,0,3],[32,0,0,0,0,9,0,0,0,0,7,24,0,0,39,34] >;

C5×C42.6C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2._6C_2^2
% in TeX

G:=Group("C5xC4^2.6C2^2");
// GroupNames label

G:=SmallGroup(320,925);
// by ID

G=gap.SmallGroup(320,925);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,1731,124]);
// Polycyclic

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

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