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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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C52C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D42Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, 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, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, Dic10, 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 52 68 99)(2 100 69 53)(3 54 70 81)(4 82 71 55)(5 56 72 83)(6 84 73 57)(7 58 74 85)(8 86 75 59)(9 60 76 87)(10 88 77 41)(11 42 78 89)(12 90 79 43)(13 44 80 91)(14 92 61 45)(15 46 62 93)(16 94 63 47)(17 48 64 95)(18 96 65 49)(19 50 66 97)(20 98 67 51)(21 134 159 117)(22 118 160 135)(23 136 141 119)(24 120 142 137)(25 138 143 101)(26 102 144 139)(27 140 145 103)(28 104 146 121)(29 122 147 105)(30 106 148 123)(31 124 149 107)(32 108 150 125)(33 126 151 109)(34 110 152 127)(35 128 153 111)(36 112 154 129)(37 130 155 113)(38 114 156 131)(39 132 157 115)(40 116 158 133)
(1 42)(2 79)(3 44)(4 61)(5 46)(6 63)(7 48)(8 65)(9 50)(10 67)(11 52)(12 69)(13 54)(14 71)(15 56)(16 73)(17 58)(18 75)(19 60)(20 77)(21 107)(22 32)(23 109)(24 34)(25 111)(26 36)(27 113)(28 38)(29 115)(30 40)(31 117)(33 119)(35 101)(37 103)(39 105)(41 51)(43 53)(45 55)(47 57)(49 59)(62 83)(64 85)(66 87)(68 89)(70 91)(72 93)(74 95)(76 97)(78 99)(80 81)(82 92)(84 94)(86 96)(88 98)(90 100)(102 129)(104 131)(106 133)(108 135)(110 137)(112 139)(114 121)(116 123)(118 125)(120 127)(122 157)(124 159)(126 141)(128 143)(130 145)(132 147)(134 149)(136 151)(138 153)(140 155)(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 39 11 29)(2 38 12 28)(3 37 13 27)(4 36 14 26)(5 35 15 25)(6 34 16 24)(7 33 17 23)(8 32 18 22)(9 31 19 21)(10 30 20 40)(41 123 51 133)(42 122 52 132)(43 121 53 131)(44 140 54 130)(45 139 55 129)(46 138 56 128)(47 137 57 127)(48 136 58 126)(49 135 59 125)(50 134 60 124)(61 144 71 154)(62 143 72 153)(63 142 73 152)(64 141 74 151)(65 160 75 150)(66 159 76 149)(67 158 77 148)(68 157 78 147)(69 156 79 146)(70 155 80 145)(81 113 91 103)(82 112 92 102)(83 111 93 101)(84 110 94 120)(85 109 95 119)(86 108 96 118)(87 107 97 117)(88 106 98 116)(89 105 99 115)(90 104 100 114)

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

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

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

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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