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## G = D4⋊Dic10order 320 = 26·5

### 2nd semidirect product of D4 and Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4⋊Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — D4×Dic5 — D4⋊Dic10
 Lower central C5 — C10 — C2×C20 — D4⋊Dic10
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D4⋊Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 422 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×2], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20, C2×C10, C2×C10 [×4], D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8 [×2], C4×D4, C4⋊Q8, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, D42Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C22×Dic5, D4×C10, C20.Q8, C20.8Q8, C406C4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, D4×Dic5, D4⋊Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, SD16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×SD16, C8⋊C22, Dic10 [×2], C22×D5, D42Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D8⋊D5, D5×SD16, D4⋊Dic10

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

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 2 2 8 10 10 20 20 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + - + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 SD16 C4○D4 D10 D10 D10 Dic10 C8⋊C22 D4⋊2D5 D4×D5 D8⋊D5 D5×SD16 kernel D4⋊Dic10 C20.Q8 C20.8Q8 C40⋊6C4 D4⋊Dic5 C5×D4⋊C4 C20⋊Q8 D4×Dic5 C2×Dic5 C5×D4 D4⋊C4 Dic5 C20 C4⋊C4 C2×C8 C2×D4 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 2 4 4

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

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 40 0
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
,
 9 11 0 0 30 14 0 0 0 0 26 26 0 0 26 15
,
 22 22 0 0 32 19 0 0 0 0 0 40 0 0 1 0
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[9,30,0,0,11,14,0,0,0,0,26,26,0,0,26,15],[22,32,0,0,22,19,0,0,0,0,0,1,0,0,40,0] >;

D4⋊Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,388);
// by ID

G=gap.SmallGroup(320,388);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,254,219,226,851,438,102,12550]);
// 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=a^-1*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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