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## G = Dic10⋊21D4order 320 = 26·5

### 9th semidirect product of Dic10 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊21D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C4×C5⋊D4 — Dic10⋊21D4
 Lower central C5 — C2×C10 — Dic10⋊21D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for Dic1021D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a9, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 934 in 280 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×15], C22, C22 [×2], C22 [×6], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×4], Q8 [×16], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×14], Dic5 [×6], Dic5 [×4], C20 [×2], C20 [×5], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C4⋊Q8 [×3], C22×Q8 [×2], Dic10 [×4], Dic10 [×10], C4×D5 [×6], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×C10, D4×Q8, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×Dic10 [×4], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4, C22×C20, Q8×C10, Dic5.14D4 [×2], Dic54D4 [×2], Dic53Q8, C20⋊Q8 [×2], D10⋊Q8 [×2], D102Q8, C4×C5⋊D4, Dic5⋊Q8, C5×C22⋊Q8, C22×Dic10, C2×Q8×D5, Dic1021D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C22×D4, C22×Q8, 2- 1+4, C22×D5 [×7], D4×Q8, D4×D5 [×2], Q8×D5 [×2], C23×D5, C2×D4×D5, C2×Q8×D5, D4.10D10, Dic1021D4

Smallest permutation representation of Dic1021D4
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 85 11 95)(2 84 12 94)(3 83 13 93)(4 82 14 92)(5 81 15 91)(6 100 16 90)(7 99 17 89)(8 98 18 88)(9 97 19 87)(10 96 20 86)(21 135 31 125)(22 134 32 124)(23 133 33 123)(24 132 34 122)(25 131 35 121)(26 130 36 140)(27 129 37 139)(28 128 38 138)(29 127 39 137)(30 126 40 136)(41 115 51 105)(42 114 52 104)(43 113 53 103)(44 112 54 102)(45 111 55 101)(46 110 56 120)(47 109 57 119)(48 108 58 118)(49 107 59 117)(50 106 60 116)(61 141 71 151)(62 160 72 150)(63 159 73 149)(64 158 74 148)(65 157 75 147)(66 156 76 146)(67 155 77 145)(68 154 78 144)(69 153 79 143)(70 152 80 142)
(1 140 61 49)(2 129 62 58)(3 138 63 47)(4 127 64 56)(5 136 65 45)(6 125 66 54)(7 134 67 43)(8 123 68 52)(9 132 69 41)(10 121 70 50)(11 130 71 59)(12 139 72 48)(13 128 73 57)(14 137 74 46)(15 126 75 55)(16 135 76 44)(17 124 77 53)(18 133 78 42)(19 122 79 51)(20 131 80 60)(21 156 102 100)(22 145 103 89)(23 154 104 98)(24 143 105 87)(25 152 106 96)(26 141 107 85)(27 150 108 94)(28 159 109 83)(29 148 110 92)(30 157 111 81)(31 146 112 90)(32 155 113 99)(33 144 114 88)(34 153 115 97)(35 142 116 86)(36 151 117 95)(37 160 118 84)(38 149 119 93)(39 158 120 82)(40 147 101 91)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 156)(22 157)(23 158)(24 159)(25 160)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(61 140)(62 121)(63 122)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 103)(82 104)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 101)(100 102)

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C ··· 4G 4H ··· 4M 4N 4O 4P 4Q 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 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 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 2 2 4 ··· 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 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 Q8 D5 D10 D10 D10 D10 2- 1+4 D4×D5 Q8×D5 D4.10D10 kernel Dic10⋊21D4 Dic5.14D4 Dic5⋊4D4 Dic5⋊3Q8 C20⋊Q8 D10⋊Q8 D10⋊2Q8 C4×C5⋊D4 Dic5⋊Q8 C5×C22⋊Q8 C22×Dic10 C2×Q8×D5 Dic10 C5⋊D4 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C4 C22 C2 # reps 1 2 2 1 2 2 1 1 1 1 1 1 4 4 2 4 6 2 2 1 4 4 4

Matrix representation of Dic1021D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 5 35 0 0 0 0 0 0 1 30 0 0 0 0 30 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 6 1 0 0 0 0 6 35 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 9 5 0 0 0 0 0 32 0 0 0 0 0 0 35 40 0 0 0 0 35 6 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 5 0 0 0 0 25 32 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

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,30,0,0,0,0,30,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[9,0,0,0,0,0,5,32,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,25,0,0,0,0,5,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

Dic1021D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{21}D_4
% in TeX

G:=Group("Dic10:21D4");
// GroupNames label

G:=SmallGroup(320,1304);
// by ID

G=gap.SmallGroup(320,1304);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,570,185,80,12550]);
// 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^9,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
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