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## G = D4×Dic10order 320 = 26·5

### Direct product of D4 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D4×Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5 — D4×Dic10
 Lower central C5 — C2×C10 — D4×Dic10
 Upper central C1 — C22 — C4×D4

Generators and relations for D4×Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 886 in 280 conjugacy classes, 123 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×13], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×20], D4 [×4], Q8 [×16], C23 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×11], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×15], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], Dic10 [×4], Dic10 [×12], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], D4×Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×6], C2×Dic10 [×8], C22×Dic5 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C202Q8, Dic5.14D4 [×4], C20⋊Q8 [×2], C20.48D4 [×2], D4×Dic5 [×2], D4×C20, C22×Dic10 [×2], D4×Dic10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C22×D4, C22×Q8, 2- 1+4, Dic10 [×4], C22×D5 [×7], D4×Q8, C2×Dic10 [×6], D4×D5 [×2], C23×D5, C22×Dic10, C2×D4×D5, D4.10D10, D4×Dic10

Smallest permutation representation of D4×Dic10
On 160 points
Generators in S160
(1 158 104 29)(2 159 105 30)(3 160 106 31)(4 141 107 32)(5 142 108 33)(6 143 109 34)(7 144 110 35)(8 145 111 36)(9 146 112 37)(10 147 113 38)(11 148 114 39)(12 149 115 40)(13 150 116 21)(14 151 117 22)(15 152 118 23)(16 153 119 24)(17 154 120 25)(18 155 101 26)(19 156 102 27)(20 157 103 28)(41 67 94 135)(42 68 95 136)(43 69 96 137)(44 70 97 138)(45 71 98 139)(46 72 99 140)(47 73 100 121)(48 74 81 122)(49 75 82 123)(50 76 83 124)(51 77 84 125)(52 78 85 126)(53 79 86 127)(54 80 87 128)(55 61 88 129)(56 62 89 130)(57 63 90 131)(58 64 91 132)(59 65 92 133)(60 66 93 134)
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(41 77)(42 78)(43 79)(44 80)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(81 132)(82 133)(83 134)(84 135)(85 136)(86 137)(87 138)(88 139)(89 140)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 145)(102 146)(103 147)(104 148)(105 149)(106 150)(107 151)(108 152)(109 153)(110 154)(111 155)(112 156)(113 157)(114 158)(115 159)(116 160)(117 141)(118 142)(119 143)(120 144)
(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)
(1 130 11 140)(2 129 12 139)(3 128 13 138)(4 127 14 137)(5 126 15 136)(6 125 16 135)(7 124 17 134)(8 123 18 133)(9 122 19 132)(10 121 20 131)(21 97 31 87)(22 96 32 86)(23 95 33 85)(24 94 34 84)(25 93 35 83)(26 92 36 82)(27 91 37 81)(28 90 38 100)(29 89 39 99)(30 88 40 98)(41 143 51 153)(42 142 52 152)(43 141 53 151)(44 160 54 150)(45 159 55 149)(46 158 56 148)(47 157 57 147)(48 156 58 146)(49 155 59 145)(50 154 60 144)(61 115 71 105)(62 114 72 104)(63 113 73 103)(64 112 74 102)(65 111 75 101)(66 110 76 120)(67 109 77 119)(68 108 78 118)(69 107 79 117)(70 106 80 116)

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 D10 D10 D10 D10 D10 Dic10 2- 1+4 D4×D5 D4.10D10 kernel D4×Dic10 C4×Dic10 C20⋊2Q8 Dic5.14D4 C20⋊Q8 C20.48D4 D4×Dic5 D4×C20 C22×Dic10 Dic10 C5×D4 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C10 C4 C2 # reps 1 1 1 4 2 2 2 1 2 4 4 2 2 4 2 4 2 16 1 4 4

Matrix representation of D4×Dic10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 9 0 0 18 1
,
 1 0 0 0 0 1 0 0 0 0 40 9 0 0 0 1
,
 9 11 0 0 30 14 0 0 0 0 1 0 0 0 0 1
,
 1 24 0 0 17 40 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,18,0,0,9,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,9,1],[9,30,0,0,11,14,0,0,0,0,1,0,0,0,0,1],[1,17,0,0,24,40,0,0,0,0,40,0,0,0,0,40] >;

D4×Dic10 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_{10}
% in TeX

G:=Group("D4xDic10");
// GroupNames label

G:=SmallGroup(320,1209);
// by ID

G=gap.SmallGroup(320,1209);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,675,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=c^10,b*a*b=a^-1,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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