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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, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, Q85D4, C4×Dic5, C4×Dic5, C10.D4, C10.D4, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C4○D20, Q8×D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, Q8×C10, D10⋊D4, Dic5.5D4, Dic53Q8, D208C4, C4⋊D20, D10⋊Q8, C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C2×C4○D20, C2×Q8×D5, Dic1022D4
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, 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 49 11 59)(2 48 12 58)(3 47 13 57)(4 46 14 56)(5 45 15 55)(6 44 16 54)(7 43 17 53)(8 42 18 52)(9 41 19 51)(10 60 20 50)(21 139 31 129)(22 138 32 128)(23 137 33 127)(24 136 34 126)(25 135 35 125)(26 134 36 124)(27 133 37 123)(28 132 38 122)(29 131 39 121)(30 130 40 140)(61 101 71 111)(62 120 72 110)(63 119 73 109)(64 118 74 108)(65 117 75 107)(66 116 76 106)(67 115 77 105)(68 114 78 104)(69 113 79 103)(70 112 80 102)(81 147 91 157)(82 146 92 156)(83 145 93 155)(84 144 94 154)(85 143 95 153)(86 142 96 152)(87 141 97 151)(88 160 98 150)(89 159 99 149)(90 158 100 148)
(1 86 127 112)(2 95 128 101)(3 84 129 110)(4 93 130 119)(5 82 131 108)(6 91 132 117)(7 100 133 106)(8 89 134 115)(9 98 135 104)(10 87 136 113)(11 96 137 102)(12 85 138 111)(13 94 139 120)(14 83 140 109)(15 92 121 118)(16 81 122 107)(17 90 123 116)(18 99 124 105)(19 88 125 114)(20 97 126 103)(21 72 47 154)(22 61 48 143)(23 70 49 152)(24 79 50 141)(25 68 51 150)(26 77 52 159)(27 66 53 148)(28 75 54 157)(29 64 55 146)(30 73 56 155)(31 62 57 144)(32 71 58 153)(33 80 59 142)(34 69 60 151)(35 78 41 160)(36 67 42 149)(37 76 43 158)(38 65 44 147)(39 74 45 156)(40 63 46 145)
(1 70)(2 79)(3 68)(4 77)(5 66)(6 75)(7 64)(8 73)(9 62)(10 71)(11 80)(12 69)(13 78)(14 67)(15 76)(16 65)(17 74)(18 63)(19 72)(20 61)(21 88)(22 97)(23 86)(24 95)(25 84)(26 93)(27 82)(28 91)(29 100)(30 89)(31 98)(32 87)(33 96)(34 85)(35 94)(36 83)(37 92)(38 81)(39 90)(40 99)(41 120)(42 109)(43 118)(44 107)(45 116)(46 105)(47 114)(48 103)(49 112)(50 101)(51 110)(52 119)(53 108)(54 117)(55 106)(56 115)(57 104)(58 113)(59 102)(60 111)(121 158)(122 147)(123 156)(124 145)(125 154)(126 143)(127 152)(128 141)(129 150)(130 159)(131 148)(132 157)(133 146)(134 155)(135 144)(136 153)(137 142)(138 151)(139 160)(140 149)

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

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

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

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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