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## G = Dic10⋊22D4order 320 = 26·5

### 10th semidirect product of Dic10 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊22D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — Dic10⋊22D4
 Lower central C5 — C2×C10 — Dic10⋊22D4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for Dic1022D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=dad=a9, cbc-1=a10b, bd=db, dcd=c-1 >

Subgroups: 1078 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×13], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], D4 [×12], Q8 [×10], C23, C23 [×3], D5 [×4], C10 [×3], C10, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], Dic5 [×3], C20 [×2], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C10 [×3], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8, C22⋊Q8 [×2], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×4], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×C10, Q85D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], D10⋊C4, D10⋊C4 [×6], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], C4○D20 [×4], Q8×D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, Q8×C10, D10⋊D4 [×2], Dic5.5D4 [×2], Dic53Q8, D208C4 [×2], C4⋊D20, D10⋊Q8 [×2], C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C2×C4○D20, C2×Q8×D5, Dic1022D4
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], C23×D5, C2×D4×D5, Q8.10D10, D5×C4○D4, Dic1022D4

Smallest permutation representation of Dic1022D4
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)
(1 121 11 131)(2 140 12 130)(3 139 13 129)(4 138 14 128)(5 137 15 127)(6 136 16 126)(7 135 17 125)(8 134 18 124)(9 133 19 123)(10 132 20 122)(21 105 31 115)(22 104 32 114)(23 103 33 113)(24 102 34 112)(25 101 35 111)(26 120 36 110)(27 119 37 109)(28 118 38 108)(29 117 39 107)(30 116 40 106)(41 72 51 62)(42 71 52 61)(43 70 53 80)(44 69 54 79)(45 68 55 78)(46 67 56 77)(47 66 57 76)(48 65 58 75)(49 64 59 74)(50 63 60 73)(81 142 91 152)(82 141 92 151)(83 160 93 150)(84 159 94 149)(85 158 95 148)(86 157 96 147)(87 156 97 146)(88 155 98 145)(89 154 99 144)(90 153 100 143)
(1 28 69 98)(2 37 70 87)(3 26 71 96)(4 35 72 85)(5 24 73 94)(6 33 74 83)(7 22 75 92)(8 31 76 81)(9 40 77 90)(10 29 78 99)(11 38 79 88)(12 27 80 97)(13 36 61 86)(14 25 62 95)(15 34 63 84)(16 23 64 93)(17 32 65 82)(18 21 66 91)(19 30 67 100)(20 39 68 89)(41 158 128 111)(42 147 129 120)(43 156 130 109)(44 145 131 118)(45 154 132 107)(46 143 133 116)(47 152 134 105)(48 141 135 114)(49 150 136 103)(50 159 137 112)(51 148 138 101)(52 157 139 110)(53 146 140 119)(54 155 121 108)(55 144 122 117)(56 153 123 106)(57 142 124 115)(58 151 125 104)(59 160 126 113)(60 149 127 102)
(1 155)(2 144)(3 153)(4 142)(5 151)(6 160)(7 149)(8 158)(9 147)(10 156)(11 145)(12 154)(13 143)(14 152)(15 141)(16 150)(17 159)(18 148)(19 157)(20 146)(21 51)(22 60)(23 49)(24 58)(25 47)(26 56)(27 45)(28 54)(29 43)(30 52)(31 41)(32 50)(33 59)(34 48)(35 57)(36 46)(37 55)(38 44)(39 53)(40 42)(61 116)(62 105)(63 114)(64 103)(65 112)(66 101)(67 110)(68 119)(69 108)(70 117)(71 106)(72 115)(73 104)(74 113)(75 102)(76 111)(77 120)(78 109)(79 118)(80 107)(81 128)(82 137)(83 126)(84 135)(85 124)(86 133)(87 122)(88 131)(89 140)(90 129)(91 138)(92 127)(93 136)(94 125)(95 134)(96 123)(97 132)(98 121)(99 130)(100 139)

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 4O 4P 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P 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 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 10 10 20 20 2 2 2 2 4 4 4 4 10 ··· 10 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 D5 C4○D4 D10 D10 D10 D10 2- 1+4 D4×D5 Q8.10D10 D5×C4○D4 kernel Dic10⋊22D4 D10⋊D4 Dic5.5D4 Dic5⋊3Q8 D20⋊8C4 C4⋊D20 D10⋊Q8 C4×C5⋊D4 C20.23D4 C5×C22⋊Q8 C2×C4○D20 C2×Q8×D5 Dic10 C22⋊Q8 D10 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C4 C2 C2 # reps 1 2 2 1 2 1 2 1 1 1 1 1 4 2 4 4 6 2 2 1 4 4 4

Matrix representation of Dic1022D4 in GL6(𝔽41)

 0 9 0 0 0 0 9 0 0 0 0 0 0 0 6 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 35 6 0 0 0 0 1 6 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 6 35 0 0 0 0 40 35 0 0 0 0 0 0 1 23 0 0 0 0 32 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 6 0 0 0 0 1 6 0 0 0 0 0 0 40 18 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,1,32,0,0,0,0,23,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,40,0,0,0,0,0,18,1] >;

Dic1022D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{22}D_4
% in TeX

G:=Group("Dic10:22D4");
// GroupNames label

G:=SmallGroup(320,1305);
// by ID

G=gap.SmallGroup(320,1305);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,12550]);
// 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=d*a*d=a^9,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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