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## G = Dic10⋊11D4order 320 = 26·5

### 4th semidirect product of Dic10 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊11D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C4×D20 — Dic10⋊11D4
 Lower central C5 — C2×C10 — Dic10⋊11D4
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for Dic1011D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 1158 in 312 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×18], C5, C2×C4 [×3], C2×C4 [×18], D4 [×24], Q8 [×4], C23 [×4], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×6], C2×D4 [×9], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], C20, D10 [×6], C2×C10, C2×C10 [×12], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C41D4, C41D4 [×2], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×8], C5⋊D4 [×12], C2×C20 [×3], C5×D4 [×10], C22×D5 [×2], C22×C10 [×4], Q86D4, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C2×Dic10, C2×C4×D5 [×2], C2×D20, D42D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×8], D4×C10 [×6], C4×Dic10, C4×D20, D4×Dic5 [×2], C202D4 [×2], Dic5⋊D4 [×4], C20⋊D4 [×2], C5×C41D4, C2×D42D5 [×2], Dic1011D4
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], D42D5 [×2], C23×D5, C2×D4×D5, C2×D42D5, D46D10, Dic1011D4

Smallest permutation representation of Dic1011D4
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)
(1 127 11 137)(2 126 12 136)(3 125 13 135)(4 124 14 134)(5 123 15 133)(6 122 16 132)(7 121 17 131)(8 140 18 130)(9 139 19 129)(10 138 20 128)(21 149 31 159)(22 148 32 158)(23 147 33 157)(24 146 34 156)(25 145 35 155)(26 144 36 154)(27 143 37 153)(28 142 38 152)(29 141 39 151)(30 160 40 150)(41 65 51 75)(42 64 52 74)(43 63 53 73)(44 62 54 72)(45 61 55 71)(46 80 56 70)(47 79 57 69)(48 78 58 68)(49 77 59 67)(50 76 60 66)(81 107 91 117)(82 106 92 116)(83 105 93 115)(84 104 94 114)(85 103 95 113)(86 102 96 112)(87 101 97 111)(88 120 98 110)(89 119 99 109)(90 118 100 108)
(1 149 64 115)(2 150 65 116)(3 151 66 117)(4 152 67 118)(5 153 68 119)(6 154 69 120)(7 155 70 101)(8 156 71 102)(9 157 72 103)(10 158 73 104)(11 159 74 105)(12 160 75 106)(13 141 76 107)(14 142 77 108)(15 143 78 109)(16 144 79 110)(17 145 80 111)(18 146 61 112)(19 147 62 113)(20 148 63 114)(21 42 93 137)(22 43 94 138)(23 44 95 139)(24 45 96 140)(25 46 97 121)(26 47 98 122)(27 48 99 123)(28 49 100 124)(29 50 81 125)(30 51 82 126)(31 52 83 127)(32 53 84 128)(33 54 85 129)(34 55 86 130)(35 56 87 131)(36 57 88 132)(37 58 89 133)(38 59 90 134)(39 60 91 135)(40 41 92 136)
(1 83)(2 94)(3 85)(4 96)(5 87)(6 98)(7 89)(8 100)(9 91)(10 82)(11 93)(12 84)(13 95)(14 86)(15 97)(16 88)(17 99)(18 90)(19 81)(20 92)(21 74)(22 65)(23 76)(24 67)(25 78)(26 69)(27 80)(28 71)(29 62)(30 73)(31 64)(32 75)(33 66)(34 77)(35 68)(36 79)(37 70)(38 61)(39 72)(40 63)(41 148)(42 159)(43 150)(44 141)(45 152)(46 143)(47 154)(48 145)(49 156)(50 147)(51 158)(52 149)(53 160)(54 151)(55 142)(56 153)(57 144)(58 155)(59 146)(60 157)(101 133)(102 124)(103 135)(104 126)(105 137)(106 128)(107 139)(108 130)(109 121)(110 132)(111 123)(112 134)(113 125)(114 136)(115 127)(116 138)(117 129)(118 140)(119 131)(120 122)

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 2+ 1+4 D4×D5 D4⋊2D5 D4⋊6D10 kernel Dic10⋊11D4 C4×Dic10 C4×D20 D4×Dic5 C20⋊2D4 Dic5⋊D4 C20⋊D4 C5×C4⋊1D4 C2×D4⋊2D5 Dic10 C4⋊1D4 C20 C42 C2×D4 C10 C4 C4 C2 # reps 1 1 1 2 2 4 2 1 2 4 2 4 2 12 1 4 4 4

Matrix representation of Dic1011D4 in GL6(𝔽41)

 1 40 0 0 0 0 36 6 0 0 0 0 0 0 32 0 0 0 0 0 32 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 35 40 0 0 0 0 35 6 0 0 0 0 0 0 40 2 0 0 0 0 40 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 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 5 0 0 0 0 16 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 2 0 0 0 0 0 1 0 0 0 0 0 0 1 36 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [1,36,0,0,0,0,40,6,0,0,0,0,0,0,32,32,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,35,0,0,0,0,40,6,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,16,0,0,0,0,5,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,36,40] >;

Dic1011D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{11}D_4
% in TeX

G:=Group("Dic10:11D4");
// GroupNames label

G:=SmallGroup(320,1390);
// by ID

G=gap.SmallGroup(320,1390);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,570,185,12550]);
// Polycyclic

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

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