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## G = D4⋊5Dic10order 320 = 26·5

### 1st semidirect product of D4 and Dic10 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D4⋊5Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5 — D4⋊5Dic10
 Lower central C5 — C2×C10 — D4⋊5Dic10
 Upper central C1 — C22 — C4×D4

Generators and relations for D45Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >

Subgroups: 694 in 228 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×4], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×C10 [×2], D43Q8, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×8], C4⋊Dic5 [×3], C4⋊Dic5 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C22×Dic5 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C20.6Q8, Dic5.14D4 [×4], C20⋊Q8, C4.Dic10, C2×C10.D4 [×2], C20.48D4 [×2], D4×Dic5 [×2], D4×C20, D45Dic10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ 1+4, Dic10 [×4], C22×D5 [×7], D43Q8, C2×Dic10 [×6], C23×D5, C22×Dic10, D46D10, D5×C4○D4, D45Dic10

Smallest permutation representation of D45Dic10
On 160 points
Generators in S160
(1 24 140 94)(2 95 121 25)(3 26 122 96)(4 97 123 27)(5 28 124 98)(6 99 125 29)(7 30 126 100)(8 81 127 31)(9 32 128 82)(10 83 129 33)(11 34 130 84)(12 85 131 35)(13 36 132 86)(14 87 133 37)(15 38 134 88)(16 89 135 39)(17 40 136 90)(18 91 137 21)(19 22 138 92)(20 93 139 23)(41 62 108 159)(42 160 109 63)(43 64 110 141)(44 142 111 65)(45 66 112 143)(46 144 113 67)(47 68 114 145)(48 146 115 69)(49 70 116 147)(50 148 117 71)(51 72 118 149)(52 150 119 73)(53 74 120 151)(54 152 101 75)(55 76 102 153)(56 154 103 77)(57 78 104 155)(58 156 105 79)(59 80 106 157)(60 158 107 61)
(1 34)(2 85)(3 36)(4 87)(5 38)(6 89)(7 40)(8 91)(9 22)(10 93)(11 24)(12 95)(13 26)(14 97)(15 28)(16 99)(17 30)(18 81)(19 32)(20 83)(21 127)(23 129)(25 131)(27 133)(29 135)(31 137)(33 139)(35 121)(37 123)(39 125)(41 149)(42 73)(43 151)(44 75)(45 153)(46 77)(47 155)(48 79)(49 157)(50 61)(51 159)(52 63)(53 141)(54 65)(55 143)(56 67)(57 145)(58 69)(59 147)(60 71)(62 118)(64 120)(66 102)(68 104)(70 106)(72 108)(74 110)(76 112)(78 114)(80 116)(82 138)(84 140)(86 122)(88 124)(90 126)(92 128)(94 130)(96 132)(98 134)(100 136)(101 142)(103 144)(105 146)(107 148)(109 150)(111 152)(113 154)(115 156)(117 158)(119 160)
(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 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 151 31 141)(22 150 32 160)(23 149 33 159)(24 148 34 158)(25 147 35 157)(26 146 36 156)(27 145 37 155)(28 144 38 154)(29 143 39 153)(30 142 40 152)(61 94 71 84)(62 93 72 83)(63 92 73 82)(64 91 74 81)(65 90 75 100)(66 89 76 99)(67 88 77 98)(68 87 78 97)(69 86 79 96)(70 85 80 95)(101 126 111 136)(102 125 112 135)(103 124 113 134)(104 123 114 133)(105 122 115 132)(106 121 116 131)(107 140 117 130)(108 139 118 129)(109 138 119 128)(110 137 120 127)

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

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

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

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 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 D10 D10 D10 Dic10 2+ 1+4 D4⋊6D10 D5×C4○D4 kernel D4⋊5Dic10 C4×Dic10 C20.6Q8 Dic5.14D4 C20⋊Q8 C4.Dic10 C2×C10.D4 C20.48D4 D4×Dic5 D4×C20 C5×D4 C4×D4 Dic5 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C10 C2 C2 # reps 1 1 1 4 1 1 2 2 2 1 4 2 4 2 4 2 4 2 16 1 4 4

Matrix representation of D45Dic10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 21 0 0 0 0 37 40
,
 1 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 21 0 0 0 0 0 40
,
 34 31 0 0 0 0 5 7 0 0 0 0 0 0 1 40 0 0 0 0 8 34 0 0 0 0 0 0 32 0 0 0 0 0 36 9
,
 40 11 0 0 0 0 11 1 0 0 0 0 0 0 3 5 0 0 0 0 23 38 0 0 0 0 0 0 1 21 0 0 0 0 37 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[34,5,0,0,0,0,31,7,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,32,36,0,0,0,0,0,9],[40,11,0,0,0,0,11,1,0,0,0,0,0,0,3,23,0,0,0,0,5,38,0,0,0,0,0,0,1,37,0,0,0,0,21,40] >;

D45Dic10 in GAP, Magma, Sage, TeX

D_4\rtimes_5{\rm Dic}_{10}
% in TeX

G:=Group("D4:5Dic10");
// GroupNames label

G:=SmallGroup(320,1211);
// by ID

G=gap.SmallGroup(320,1211);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,675,192,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=c^10,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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