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## G = Dic10⋊20D4order 320 = 26·5

### 8th semidirect product of Dic10 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊20D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C2×C4○D20 — Dic10⋊20D4
 Lower central C5 — C2×C10 — Dic10⋊20D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for Dic1020D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1222 in 312 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×18], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×24], Q8 [×4], C23, C23 [×2], C23 [×3], D5 [×3], C10 [×3], C10 [×3], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×12], C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×2], C20 [×2], C20 [×3], D10 [×9], C2×C10, C2×C10 [×9], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×5], C41D4 [×3], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×6], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×2], C22×C10, C22×C10 [×2], Q86D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C4○D20 [×4], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4 [×8], C22×C20, D4×C10, D4×C10 [×2], Dic54D4 [×2], D10⋊D4 [×2], Dic53Q8, C4⋊D20, C4×C5⋊D4, Dic5⋊D4 [×2], C20⋊D4, C20⋊D4 [×2], C5×C4⋊D4, C2×C4○D20, C2×D42D5, Dic1020D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C22×D5 [×7], Q86D4, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10, D5×C4○D4, Dic1020D4

Smallest permutation representation of Dic1020D4
On 160 points
Generators in S160
(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)
(1 115 11 105)(2 114 12 104)(3 113 13 103)(4 112 14 102)(5 111 15 101)(6 110 16 120)(7 109 17 119)(8 108 18 118)(9 107 19 117)(10 106 20 116)(21 132 31 122)(22 131 32 121)(23 130 33 140)(24 129 34 139)(25 128 35 138)(26 127 36 137)(27 126 37 136)(28 125 38 135)(29 124 39 134)(30 123 40 133)(41 158 51 148)(42 157 52 147)(43 156 53 146)(44 155 54 145)(45 154 55 144)(46 153 56 143)(47 152 57 142)(48 151 58 141)(49 150 59 160)(50 149 60 159)(61 100 71 90)(62 99 72 89)(63 98 73 88)(64 97 74 87)(65 96 75 86)(66 95 76 85)(67 94 77 84)(68 93 78 83)(69 92 79 82)(70 91 80 81)
(1 91 39 52)(2 82 40 43)(3 93 21 54)(4 84 22 45)(5 95 23 56)(6 86 24 47)(7 97 25 58)(8 88 26 49)(9 99 27 60)(10 90 28 51)(11 81 29 42)(12 92 30 53)(13 83 31 44)(14 94 32 55)(15 85 33 46)(16 96 34 57)(17 87 35 48)(18 98 36 59)(19 89 37 50)(20 100 38 41)(61 135 148 116)(62 126 149 107)(63 137 150 118)(64 128 151 109)(65 139 152 120)(66 130 153 111)(67 121 154 102)(68 132 155 113)(69 123 156 104)(70 134 157 115)(71 125 158 106)(72 136 159 117)(73 127 160 108)(74 138 141 119)(75 129 142 110)(76 140 143 101)(77 131 144 112)(78 122 145 103)(79 133 146 114)(80 124 147 105)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 61)(59 62)(60 63)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 141)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 150)(100 151)

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 5A 5B 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 20A ··· 20H 20I 20J 20K 20L order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 5 5 10 ··· 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 4 20 20 20 2 2 2 2 4 4 10 ··· 10 20 2 2 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4 8 8 8 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4×D5 D4⋊6D10 D5×C4○D4 kernel Dic10⋊20D4 Dic5⋊4D4 D10⋊D4 Dic5⋊3Q8 C4⋊D20 C4×C5⋊D4 Dic5⋊D4 C20⋊D4 C5×C4⋊D4 C2×C4○D20 C2×D4⋊2D5 Dic10 C4⋊D4 Dic5 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C4 C2 C2 # reps 1 2 2 1 1 1 2 3 1 1 1 4 2 4 4 2 2 6 1 4 4 4

Matrix representation of Dic1020D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 22 2 0 0 0 0 24 19 0 0 0 0 0 0 7 40 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 7 32 0 0 0 0 0 0 7 34 0 0 0 0 1 34
,
 1 40 0 0 0 0 2 40 0 0 0 0 0 0 7 23 0 0 0 0 21 34 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 2 40 0 0 0 0 0 0 7 23 0 0 0 0 30 34 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,24,0,0,0,0,2,19,0,0,0,0,0,0,7,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,7,0,0,0,0,0,32,0,0,0,0,0,0,7,1,0,0,0,0,34,34],[1,2,0,0,0,0,40,40,0,0,0,0,0,0,7,21,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,7,30,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic1020D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{20}D_4
% in TeX

G:=Group("Dic10:20D4");
// GroupNames label

G:=SmallGroup(320,1271);
// by ID

G=gap.SmallGroup(320,1271);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,184,570,185,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

׿
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