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## G = Dic10⋊19D4order 320 = 26·5

### 7th semidirect product of Dic10 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊19D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C22×Dic10 — Dic10⋊19D4
 Lower central C5 — C2×C10 — Dic10⋊19D4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for Dic1019D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 982 in 290 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×19], D4 [×12], Q8 [×10], C23, C23 [×2], C23, D5, C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×8], C4○D4 [×4], Dic5 [×4], Dic5 [×5], C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×6], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×8], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×2], Q85D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×C4×D5, D42D5 [×4], C22×Dic5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], Dic5.5D4 [×2], Dic53Q8, D102Q8, C4×C5⋊D4, D4×Dic5 [×2], C20.17D4, Dic5⋊D4 [×2], C5×C4⋊D4, C22×Dic10, C2×D42D5, Dic1019D4
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], Q85D4, D4×D5 [×2], D42D5 [×2], C23×D5, C2×D4×D5, C2×D42D5, D4.10D10, Dic1019D4

Smallest permutation representation of Dic1019D4
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 83 11 93)(2 82 12 92)(3 81 13 91)(4 100 14 90)(5 99 15 89)(6 98 16 88)(7 97 17 87)(8 96 18 86)(9 95 19 85)(10 94 20 84)(21 80 31 70)(22 79 32 69)(23 78 33 68)(24 77 34 67)(25 76 35 66)(26 75 36 65)(27 74 37 64)(28 73 38 63)(29 72 39 62)(30 71 40 61)(41 110 51 120)(42 109 52 119)(43 108 53 118)(44 107 54 117)(45 106 55 116)(46 105 56 115)(47 104 57 114)(48 103 58 113)(49 102 59 112)(50 101 60 111)(121 142 131 152)(122 141 132 151)(123 160 133 150)(124 159 134 149)(125 158 135 148)(126 157 136 147)(127 156 137 146)(128 155 138 145)(129 154 139 144)(130 153 140 143)
(1 36 155 52)(2 27 156 43)(3 38 157 54)(4 29 158 45)(5 40 159 56)(6 31 160 47)(7 22 141 58)(8 33 142 49)(9 24 143 60)(10 35 144 51)(11 26 145 42)(12 37 146 53)(13 28 147 44)(14 39 148 55)(15 30 149 46)(16 21 150 57)(17 32 151 48)(18 23 152 59)(19 34 153 50)(20 25 154 41)(61 134 115 99)(62 125 116 90)(63 136 117 81)(64 127 118 92)(65 138 119 83)(66 129 120 94)(67 140 101 85)(68 131 102 96)(69 122 103 87)(70 133 104 98)(71 124 105 89)(72 135 106 100)(73 126 107 91)(74 137 108 82)(75 128 109 93)(76 139 110 84)(77 130 111 95)(78 121 112 86)(79 132 113 97)(80 123 114 88)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 97)(42 98)(43 99)(44 100)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 156)(62 157)(63 158)(64 159)(65 160)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F ··· 4M 4N 4O 4P 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 4 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 2 2 4 4 20 2 2 4 4 4 10 ··· 10 20 20 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 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 C2 D4 D5 C4○D4 D10 D10 D10 D10 2- 1+4 D4×D5 D4⋊2D5 D4.10D10 kernel Dic10⋊19D4 Dic5.14D4 Dic5.5D4 Dic5⋊3Q8 D10⋊2Q8 C4×C5⋊D4 D4×Dic5 C20.17D4 Dic5⋊D4 C5×C4⋊D4 C22×Dic10 C2×D4⋊2D5 Dic10 C4⋊D4 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C4 C22 C2 # reps 1 2 2 1 1 1 2 1 2 1 1 1 4 2 4 4 2 2 6 1 4 4 4

Matrix representation of Dic1019D4 in GL6(𝔽41)

 9 0 0 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 40 34 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 9 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 23 31 0 0 0 0 12 18
,
 1 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 0 18 10 0 0 0 0 21 23

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,12,0,0,0,0,31,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,21,0,0,0,0,10,23] >;

Dic1019D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{19}D_4
% in TeX

G:=Group("Dic10:19D4");
// GroupNames label

G:=SmallGroup(320,1270);
// by ID

G=gap.SmallGroup(320,1270);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,570,185,12550]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
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