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## G = Dic10⋊9D4order 320 = 26·5

### 2nd semidirect product of Dic10 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10⋊9D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C4×Dic10 — Dic10⋊9D4
 Lower central C5 — C10 — C2×C20 — Dic10⋊9D4
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for Dic109D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a11, bc=cb, dbd=a5b, dcd=c-1 >

Subgroups: 438 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×3], C23 [×2], C10 [×3], C10 [×2], C42, C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], C20, C2×C10, C2×C10 [×6], D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C5×D4 [×8], C22×C10 [×2], C4⋊SD16, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, D4.D5 [×4], C4×C20, C2×Dic10, D4×C10 [×2], D4×C10 [×2], C203C8, D4⋊Dic5 [×2], C4×Dic10, C2×D4.D5 [×2], C5×C41D4, Dic109D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8⋊C22, C5⋊D4 [×2], C22×D5, C4⋊SD16, D4.D5 [×2], D4×D5, D42D5, C2×C5⋊D4, D4.D10, C2×D4.D5, C202D4, Dic109D4

Smallest permutation representation of Dic109D4
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 68 11 78)(2 67 12 77)(3 66 13 76)(4 65 14 75)(5 64 15 74)(6 63 16 73)(7 62 17 72)(8 61 18 71)(9 80 19 70)(10 79 20 69)(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)(41 113 51 103)(42 112 52 102)(43 111 53 101)(44 110 54 120)(45 109 55 119)(46 108 56 118)(47 107 57 117)(48 106 58 116)(49 105 59 115)(50 104 60 114)(81 126 91 136)(82 125 92 135)(83 124 93 134)(84 123 94 133)(85 122 95 132)(86 121 96 131)(87 140 97 130)(88 139 98 129)(89 138 99 128)(90 137 100 127)
(1 81 48 24)(2 82 49 25)(3 83 50 26)(4 84 51 27)(5 85 52 28)(6 86 53 29)(7 87 54 30)(8 88 55 31)(9 89 56 32)(10 90 57 33)(11 91 58 34)(12 92 59 35)(13 93 60 36)(14 94 41 37)(15 95 42 38)(16 96 43 39)(17 97 44 40)(18 98 45 21)(19 99 46 22)(20 100 47 23)(61 139 119 159)(62 140 120 160)(63 121 101 141)(64 122 102 142)(65 123 103 143)(66 124 104 144)(67 125 105 145)(68 126 106 146)(69 127 107 147)(70 128 108 148)(71 129 109 149)(72 130 110 150)(73 131 111 151)(74 132 112 152)(75 133 113 153)(76 134 114 154)(77 135 115 155)(78 136 116 156)(79 137 117 157)(80 138 118 158)
(2 12)(4 14)(6 16)(8 18)(10 20)(21 88)(22 99)(23 90)(24 81)(25 92)(26 83)(27 94)(28 85)(29 96)(30 87)(31 98)(32 89)(33 100)(34 91)(35 82)(36 93)(37 84)(38 95)(39 86)(40 97)(41 51)(43 53)(45 55)(47 57)(49 59)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)(101 106)(102 117)(103 108)(104 119)(105 110)(107 112)(109 114)(111 116)(113 118)(115 120)(121 146)(122 157)(123 148)(124 159)(125 150)(126 141)(127 152)(128 143)(129 154)(130 145)(131 156)(132 147)(133 158)(134 149)(135 160)(136 151)(137 142)(138 153)(139 144)(140 155)

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

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

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

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 ··· 10N 20A ··· 20L 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 20 ··· 20 size 1 1 1 1 8 8 2 2 2 2 4 20 20 20 20 2 2 20 20 20 20 2 ··· 2 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 D4 D4 D5 SD16 C4○D4 D10 D10 C5⋊D4 C8⋊C22 D4.D5 D4×D5 D4⋊2D5 D4.D10 kernel Dic10⋊9D4 C20⋊3C8 D4⋊Dic5 C4×Dic10 C2×D4.D5 C5×C4⋊1D4 Dic10 C2×C20 C4⋊1D4 C20 C20 C42 C2×D4 C2×C4 C10 C4 C4 C4 C2 # reps 1 1 2 1 2 1 2 2 2 4 2 2 4 8 1 4 2 2 4

Matrix representation of Dic109D4 in GL6(𝔽41)

 1 37 0 0 0 0 21 40 0 0 0 0 0 0 31 0 0 0 0 0 9 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 22 0 0 0 0 13 0 0 0 0 0 0 0 19 25 0 0 0 0 2 22 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 16 18 0 0 0 0 29 25
,
 1 0 0 0 0 0 21 40 0 0 0 0 0 0 1 0 0 0 0 0 28 40 0 0 0 0 0 0 1 0 0 0 0 0 21 40

G:=sub<GL(6,GF(41))| [1,21,0,0,0,0,37,40,0,0,0,0,0,0,31,9,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,22,0,0,0,0,0,0,0,19,2,0,0,0,0,25,22,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,29,0,0,0,0,18,25],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40] >;

Dic109D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_9D_4
% in TeX

G:=Group("Dic10:9D4");
// GroupNames label

G:=SmallGroup(320,702);
// by ID

G=gap.SmallGroup(320,702);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,1123,297,136,12550]);
// 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^5*b,d*c*d=c^-1>;
// generators/relations

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