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## G = D4⋊6Dic10order 320 = 26·5

### 2nd semidirect product of D4 and Dic10 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D4⋊6Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5 — D4⋊6Dic10
 Lower central C5 — C2×C10 — D4⋊6Dic10
 Upper central C1 — C22 — C4×D4

Generators and relations for D46Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 694 in 228 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], 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 [×8], C20 [×4], C20 [×3], 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 [×8], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], D43Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×8], 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, C202Q8, Dic5.14D4 [×4], C4.Dic10 [×2], C20.48D4 [×2], C2×C4⋊Dic5 [×2], D4×Dic5 [×2], D4×C20, D46Dic10
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], D42D5 [×2], C23×D5, C22×Dic10, C2×D42D5, D48D10, D46Dic10

Smallest permutation representation of D46Dic10
On 160 points
Generators in S160
(1 51 143 103)(2 52 144 104)(3 53 145 105)(4 54 146 106)(5 55 147 107)(6 56 148 108)(7 57 149 109)(8 58 150 110)(9 59 151 111)(10 60 152 112)(11 41 153 113)(12 42 154 114)(13 43 155 115)(14 44 156 116)(15 45 157 117)(16 46 158 118)(17 47 159 119)(18 48 160 120)(19 49 141 101)(20 50 142 102)(21 137 66 89)(22 138 67 90)(23 139 68 91)(24 140 69 92)(25 121 70 93)(26 122 71 94)(27 123 72 95)(28 124 73 96)(29 125 74 97)(30 126 75 98)(31 127 76 99)(32 128 77 100)(33 129 78 81)(34 130 79 82)(35 131 80 83)(36 132 61 84)(37 133 62 85)(38 134 63 86)(39 135 64 87)(40 136 65 88)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 99)(22 100)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 133)(73 134)(74 135)(75 136)(76 137)(77 138)(78 139)(79 140)(80 121)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)
(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 72 11 62)(2 71 12 61)(3 70 13 80)(4 69 14 79)(5 68 15 78)(6 67 16 77)(7 66 17 76)(8 65 18 75)(9 64 19 74)(10 63 20 73)(21 159 31 149)(22 158 32 148)(23 157 33 147)(24 156 34 146)(25 155 35 145)(26 154 36 144)(27 153 37 143)(28 152 38 142)(29 151 39 141)(30 150 40 160)(41 85 51 95)(42 84 52 94)(43 83 53 93)(44 82 54 92)(45 81 55 91)(46 100 56 90)(47 99 57 89)(48 98 58 88)(49 97 59 87)(50 96 60 86)(101 125 111 135)(102 124 112 134)(103 123 113 133)(104 122 114 132)(105 121 115 131)(106 140 116 130)(107 139 117 129)(108 138 118 128)(109 137 119 127)(110 136 120 126)

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

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

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

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 Q8 D5 C4○D4 D10 D10 D10 D10 D10 Dic10 2+ 1+4 D4⋊2D5 D4⋊8D10 kernel D4⋊6Dic10 C4×Dic10 C20⋊2Q8 Dic5.14D4 C4.Dic10 C20.48D4 C2×C4⋊Dic5 D4×Dic5 D4×C20 C5×D4 C4×D4 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C10 C4 C2 # reps 1 1 1 4 2 2 2 2 1 4 2 4 2 4 2 4 2 16 1 4 4

Matrix representation of D46Dic10 in GL6(𝔽41)

 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 39 0 0 0 0 1 1
,
 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 2 0 0 0 0 0 40
,
 34 40 0 0 0 0 8 1 0 0 0 0 0 0 40 5 0 0 0 0 16 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 35 12 0 0 0 0 21 6 0 0 0 0 0 0 7 12 0 0 0 0 30 34 0 0 0 0 0 0 9 18 0 0 0 0 32 32

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[34,8,0,0,0,0,40,1,0,0,0,0,0,0,40,16,0,0,0,0,5,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,21,0,0,0,0,12,6,0,0,0,0,0,0,7,30,0,0,0,0,12,34,0,0,0,0,0,0,9,32,0,0,0,0,18,32] >;

D46Dic10 in GAP, Magma, Sage, TeX

D_4\rtimes_6{\rm Dic}_{10}
% in TeX

G:=Group("D4:6Dic10");
// GroupNames label

G:=SmallGroup(320,1215);
// by ID

G=gap.SmallGroup(320,1215);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1571,192,12550]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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