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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, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, Q85D4, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, Dic5.14D4, Dic5.5D4, Dic53Q8, D102Q8, C4×C5⋊D4, D4×Dic5, C20.17D4, Dic5⋊D4, C5×C4⋊D4, C22×Dic10, C2×D42D5, Dic1019D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C22×D5, Q85D4, D4×D5, D42D5, 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)
(1 129 11 139)(2 128 12 138)(3 127 13 137)(4 126 14 136)(5 125 15 135)(6 124 16 134)(7 123 17 133)(8 122 18 132)(9 121 19 131)(10 140 20 130)(21 78 31 68)(22 77 32 67)(23 76 33 66)(24 75 34 65)(25 74 35 64)(26 73 36 63)(27 72 37 62)(28 71 38 61)(29 70 39 80)(30 69 40 79)(41 90 51 100)(42 89 52 99)(43 88 53 98)(44 87 54 97)(45 86 55 96)(46 85 56 95)(47 84 57 94)(48 83 58 93)(49 82 59 92)(50 81 60 91)(101 148 111 158)(102 147 112 157)(103 146 113 156)(104 145 114 155)(105 144 115 154)(106 143 116 153)(107 142 117 152)(108 141 118 151)(109 160 119 150)(110 159 120 149)
(1 156 90 28)(2 147 91 39)(3 158 92 30)(4 149 93 21)(5 160 94 32)(6 151 95 23)(7 142 96 34)(8 153 97 25)(9 144 98 36)(10 155 99 27)(11 146 100 38)(12 157 81 29)(13 148 82 40)(14 159 83 31)(15 150 84 22)(16 141 85 33)(17 152 86 24)(18 143 87 35)(19 154 88 26)(20 145 89 37)(41 61 139 113)(42 72 140 104)(43 63 121 115)(44 74 122 106)(45 65 123 117)(46 76 124 108)(47 67 125 119)(48 78 126 110)(49 69 127 101)(50 80 128 112)(51 71 129 103)(52 62 130 114)(53 73 131 105)(54 64 132 116)(55 75 133 107)(56 66 134 118)(57 77 135 109)(58 68 136 120)(59 79 137 111)(60 70 138 102)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 61)(17 62)(18 63)(19 64)(20 65)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 137)(32 138)(33 139)(34 140)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 151)(52 152)(53 153)(54 154)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)(99 107)(100 108)

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

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

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

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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