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## G = Dic10⋊23D4order 320 = 26·5

### 1st semidirect product of Dic10 and D4 acting through Inn(Dic10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊23D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — Dic5⋊4D4 — Dic10⋊23D4
 Lower central C5 — C2×C10 — Dic10⋊23D4
 Upper central C1 — C22 — C4×D4

Generators and relations for Dic1023D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd=c-1 >

Subgroups: 1030 in 290 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8 [×8], C4○D4 [×4], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×5], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×6], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], Q85D4, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C4.D20, Dic54D4 [×2], Dic5.5D4 [×2], D10⋊Q8 [×2], C20.48D4, C207D4, Dic5⋊D4 [×2], D4×C20, C22×Dic10, C2×C4○D20, Dic1023D4
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, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4.10D10, Dic1023D4

Smallest permutation representation of Dic1023D4
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 54 11 44)(2 53 12 43)(3 52 13 42)(4 51 14 41)(5 50 15 60)(6 49 16 59)(7 48 17 58)(8 47 18 57)(9 46 19 56)(10 45 20 55)(21 124 31 134)(22 123 32 133)(23 122 33 132)(24 121 34 131)(25 140 35 130)(26 139 36 129)(27 138 37 128)(28 137 38 127)(29 136 39 126)(30 135 40 125)(61 81 71 91)(62 100 72 90)(63 99 73 89)(64 98 74 88)(65 97 75 87)(66 96 76 86)(67 95 77 85)(68 94 78 84)(69 93 79 83)(70 92 80 82)(101 149 111 159)(102 148 112 158)(103 147 113 157)(104 146 114 156)(105 145 115 155)(106 144 116 154)(107 143 117 153)(108 142 118 152)(109 141 119 151)(110 160 120 150)
(1 156 80 28)(2 157 61 29)(3 158 62 30)(4 159 63 31)(5 160 64 32)(6 141 65 33)(7 142 66 34)(8 143 67 35)(9 144 68 36)(10 145 69 37)(11 146 70 38)(12 147 71 39)(13 148 72 40)(14 149 73 21)(15 150 74 22)(16 151 75 23)(17 152 76 24)(18 153 77 25)(19 154 78 26)(20 155 79 27)(41 101 89 134)(42 102 90 135)(43 103 91 136)(44 104 92 137)(45 105 93 138)(46 106 94 139)(47 107 95 140)(48 108 96 121)(49 109 97 122)(50 110 98 123)(51 111 99 124)(52 112 100 125)(53 113 81 126)(54 114 82 127)(55 115 83 128)(56 116 84 129)(57 117 85 130)(58 118 86 131)(59 119 87 132)(60 120 88 133)
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 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 D10 C4○D20 2- 1+4 D4×D5 D4.10D10 kernel Dic10⋊23D4 C4×Dic10 C4.D20 Dic5⋊4D4 Dic5.5D4 D10⋊Q8 C20.48D4 C20⋊7D4 Dic5⋊D4 D4×C20 C22×Dic10 C2×C4○D20 Dic10 C4×D4 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C10 C4 C2 # reps 1 1 1 2 2 2 1 1 2 1 1 1 4 2 4 2 4 2 4 2 16 1 4 4

Matrix representation of Dic1023D4 in GL4(𝔽41) generated by

 11 16 0 0 39 27 0 0 0 0 1 0 0 0 0 1
,
 38 20 0 0 20 3 0 0 0 0 40 0 0 0 0 40
,
 24 34 0 0 6 17 0 0 0 0 1 2 0 0 40 40
,
 1 0 0 0 0 1 0 0 0 0 40 39 0 0 0 1
G:=sub<GL(4,GF(41))| [11,39,0,0,16,27,0,0,0,0,1,0,0,0,0,1],[38,20,0,0,20,3,0,0,0,0,40,0,0,0,0,40],[24,6,0,0,34,17,0,0,0,0,1,40,0,0,2,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,39,1] >;

Dic1023D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{23}D_4
% in TeX

G:=Group("Dic10:23D4");
// GroupNames label

G:=SmallGroup(320,1224);
// by ID

G=gap.SmallGroup(320,1224);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,100,675,570,12550]);
// 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,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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