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## G = D5×C42.C2order 320 = 26·5

### Direct product of D5 and C42.C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×C42.C2
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — D5×C42.C2
 Lower central C5 — C2×C10 — D5×C42.C2
 Upper central C1 — C22 — C42.C2

Generators and relations for D5×C42.C2
G = < a,b,c,d,e | a5=b2=c4=d4=1, e2=d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd2, ede-1=c2d >

Subgroups: 686 in 226 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×14], C22, C22 [×6], C5, C2×C4, C2×C4 [×6], C2×C4 [×23], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C4⋊C4 [×6], C4⋊C4 [×18], C22×C4 [×7], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×6], D10 [×6], C2×C10, C2×C42, C2×C4⋊C4 [×6], C42.C2, C42.C2 [×7], C4×D5 [×4], C4×D5 [×12], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C22×D5, C2×C42.C2, C4×Dic5, C4×Dic5 [×2], C10.D4 [×12], C4⋊Dic5 [×6], C4×C20, C5×C4⋊C4 [×6], C2×C4×D5, C2×C4×D5 [×6], C20.6Q8, D5×C42, Dic5.Q8 [×4], C4.Dic10 [×2], D5×C4⋊C4 [×6], C5×C42.C2, D5×C42.C2
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C22×D5 [×7], C2×C42.C2, Q8×D5 [×2], C23×D5, C2×Q8×D5, D5×C4○D4 [×2], D5×C42.C2

Smallest permutation representation of D5×C42.C2
On 160 points
Generators in S160
(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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)(81 86)(82 90)(83 89)(84 88)(85 87)(91 96)(92 100)(93 99)(94 98)(95 97)(101 106)(102 110)(103 109)(104 108)(105 107)(111 116)(112 120)(113 119)(114 118)(115 117)(121 126)(122 130)(123 129)(124 128)(125 127)(131 136)(132 140)(133 139)(134 138)(135 137)(141 146)(142 150)(143 149)(144 148)(145 147)(151 156)(152 160)(153 159)(154 158)(155 157)
(1 89 9 84)(2 90 10 85)(3 86 6 81)(4 87 7 82)(5 88 8 83)(11 96 16 91)(12 97 17 92)(13 98 18 93)(14 99 19 94)(15 100 20 95)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(1 54 14 44)(2 55 15 45)(3 51 11 41)(4 52 12 42)(5 53 13 43)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)
(1 119 14 109)(2 120 15 110)(3 116 11 106)(4 117 12 107)(5 118 13 108)(6 111 16 101)(7 112 17 102)(8 113 18 103)(9 114 19 104)(10 115 20 105)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 151 51 141)(42 152 52 142)(43 153 53 143)(44 154 54 144)(45 155 55 145)(46 156 56 146)(47 157 57 147)(48 158 58 148)(49 159 59 149)(50 160 60 150)(61 136 71 126)(62 137 72 127)(63 138 73 128)(64 139 74 129)(65 140 75 130)(66 131 76 121)(67 132 77 122)(68 133 78 123)(69 134 79 124)(70 135 80 125)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 5 5 5 5 2 ··· 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 Q8×D5 D5×C4○D4 kernel D5×C42.C2 C20.6Q8 D5×C42 Dic5.Q8 C4.Dic10 D5×C4⋊C4 C5×C42.C2 C4×D5 C42.C2 D10 C42 C4⋊C4 C4 C2 # reps 1 1 1 4 2 6 1 4 2 8 2 12 4 8

Matrix representation of D5×C42.C2 in GL6(𝔽41)

 6 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 6 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 4 29 0 0 0 0 39 37
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 22 3 0 0 0 0 3 19 0 0 0 0 0 0 4 29 0 0 0 0 39 37
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 9 0 0 0 0 0 0 0 8 37 0 0 0 0 6 33

G:=sub<GL(6,GF(41))| [6,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,6,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,4,39,0,0,0,0,29,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,3,0,0,0,0,3,19,0,0,0,0,0,0,4,39,0,0,0,0,29,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,0,0,8,6,0,0,0,0,37,33] >;

D5×C42.C2 in GAP, Magma, Sage, TeX

D_5\times C_4^2.C_2
% in TeX

G:=Group("D5xC4^2.C2");
// GroupNames label

G:=SmallGroup(320,1359);
// by ID

G=gap.SmallGroup(320,1359);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,346,297,136,12550]);
// Polycyclic

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

׿
×
𝔽