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## G = C4×D4.D5order 320 = 26·5

### Direct product of C4 and D4.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C4×D4.D5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C2×D4.D5 — C4×D4.D5
 Lower central C5 — C10 — C20 — C4×D4.D5
 Upper central C1 — C2×C4 — C42 — C4×D4

Generators and relations for C4×D4.D5
G = < a,b,c,d,e | a4=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 358 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C52C8, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C4×SD16, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C4×C52C8, C20.Q8, C10.Q16, D4⋊Dic5, C4×Dic10, C2×D4.D5, D4×C20, C4×D4.D5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, SD16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×SD16, C4○D8, C4×D5, C5⋊D4, C22×D5, C4×SD16, D4.D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C2×D4.D5, D4.8D10, C4×D4.D5

Smallest permutation representation of C4×D4.D5
On 160 points
Generators in S160
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 141 101 121)(82 142 102 122)(83 143 103 123)(84 144 104 124)(85 145 105 125)(86 146 106 126)(87 147 107 127)(88 148 108 128)(89 149 109 129)(90 150 110 130)(91 151 111 131)(92 152 112 132)(93 153 113 133)(94 154 114 134)(95 155 115 135)(96 156 116 136)(97 157 117 137)(98 158 118 138)(99 159 119 139)(100 160 120 140)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 146)(122 147)(123 148)(124 149)(125 150)(126 141)(127 142)(128 143)(129 144)(130 145)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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 88 6 83)(2 87 7 82)(3 86 8 81)(4 90 9 85)(5 89 10 84)(11 98 16 93)(12 97 17 92)(13 96 18 91)(14 100 19 95)(15 99 20 94)(21 108 26 103)(22 107 27 102)(23 106 28 101)(24 110 29 105)(25 109 30 104)(31 118 36 113)(32 117 37 112)(33 116 38 111)(34 120 39 115)(35 119 40 114)(41 128 46 123)(42 127 47 122)(43 126 48 121)(44 130 49 125)(45 129 50 124)(51 138 56 133)(52 137 57 132)(53 136 58 131)(54 140 59 135)(55 139 60 134)(61 148 66 143)(62 147 67 142)(63 146 68 141)(64 150 69 145)(65 149 70 144)(71 158 76 153)(72 157 77 152)(73 156 78 151)(74 160 79 155)(75 159 80 154)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 8A ··· 8H 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 4 20 20 20 20 2 2 10 ··· 10 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D5 SD16 C4○D4 D10 D10 D10 C4○D8 C5⋊D4 C4×D5 C4○D20 D4.D5 D4.8D10 kernel C4×D4.D5 C4×C5⋊2C8 C20.Q8 C10.Q16 D4⋊Dic5 C4×Dic10 C2×D4.D5 D4×C20 D4.D5 C2×C20 C4×D4 C20 C20 C42 C4⋊C4 C2×D4 C10 C2×C4 D4 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 4 2 2 2 2 4 8 8 8 4 4

Matrix representation of C4×D4.D5 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 1 0 0 0 3 40 0 0 0 0 0 1 0 0 1 0
,
 16 0 0 0 38 18 0 0 0 0 1 0 0 0 0 1
,
 16 3 0 0 38 25 0 0 0 0 15 26 0 0 26 26
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,3,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[16,38,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[16,38,0,0,3,25,0,0,0,0,15,26,0,0,26,26] >;

C4×D4.D5 in GAP, Magma, Sage, TeX

C_4\times D_4.D_5
% in TeX

G:=Group("C4xD4.D5");
// GroupNames label

G:=SmallGroup(320,644);
// by ID

G=gap.SmallGroup(320,644);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,58,1684,851,102,12550]);
// Polycyclic

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

׿
×
𝔽