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## G = D4.2Dic10order 320 = 26·5

### 2nd non-split extension by D4 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.2Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — D4×Dic5 — D4.2Dic10
 Lower central C5 — C10 — C2×C20 — D4.2Dic10
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D4.2Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=a2c10, bab=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd-1=c-1 >

Subgroups: 374 in 102 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4.Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C22×Dic5, D4×C10, C20.Q8, C20.8Q8, C405C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, D4×Dic5, D4.2Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, Dic10, C22×D5, D4.Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D83D5, D40⋊C2, D4.2Dic10

Smallest permutation representation of D4.2Dic10
On 160 points
Generators in S160
(1 112 121 149)(2 150 122 113)(3 114 123 151)(4 152 124 115)(5 116 125 153)(6 154 126 117)(7 118 127 155)(8 156 128 119)(9 120 129 157)(10 158 130 101)(11 102 131 159)(12 160 132 103)(13 104 133 141)(14 142 134 105)(15 106 135 143)(16 144 136 107)(17 108 137 145)(18 146 138 109)(19 110 139 147)(20 148 140 111)(21 69 94 53)(22 54 95 70)(23 71 96 55)(24 56 97 72)(25 73 98 57)(26 58 99 74)(27 75 100 59)(28 60 81 76)(29 77 82 41)(30 42 83 78)(31 79 84 43)(32 44 85 80)(33 61 86 45)(34 46 87 62)(35 63 88 47)(36 48 89 64)(37 65 90 49)(38 50 91 66)(39 67 92 51)(40 52 93 68)
(1 102)(2 132)(3 104)(4 134)(5 106)(6 136)(7 108)(8 138)(9 110)(10 140)(11 112)(12 122)(13 114)(14 124)(15 116)(16 126)(17 118)(18 128)(19 120)(20 130)(21 43)(22 32)(23 45)(24 34)(25 47)(26 36)(27 49)(28 38)(29 51)(30 40)(31 53)(33 55)(35 57)(37 59)(39 41)(42 68)(44 70)(46 72)(48 74)(50 76)(52 78)(54 80)(56 62)(58 64)(60 66)(61 96)(63 98)(65 100)(67 82)(69 84)(71 86)(73 88)(75 90)(77 92)(79 94)(81 91)(83 93)(85 95)(87 97)(89 99)(101 111)(103 113)(105 115)(107 117)(109 119)(121 159)(123 141)(125 143)(127 145)(129 147)(131 149)(133 151)(135 153)(137 155)(139 157)(142 152)(144 154)(146 156)(148 158)(150 160)
(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 45 131 71)(2 44 132 70)(3 43 133 69)(4 42 134 68)(5 41 135 67)(6 60 136 66)(7 59 137 65)(8 58 138 64)(9 57 139 63)(10 56 140 62)(11 55 121 61)(12 54 122 80)(13 53 123 79)(14 52 124 78)(15 51 125 77)(16 50 126 76)(17 49 127 75)(18 48 128 74)(19 47 129 73)(20 46 130 72)(21 151 84 104)(22 150 85 103)(23 149 86 102)(24 148 87 101)(25 147 88 120)(26 146 89 119)(27 145 90 118)(28 144 91 117)(29 143 92 116)(30 142 93 115)(31 141 94 114)(32 160 95 113)(33 159 96 112)(34 158 97 111)(35 157 98 110)(36 156 99 109)(37 155 100 108)(38 154 81 107)(39 153 82 106)(40 152 83 105)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 2 2 8 10 10 20 20 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + - + - + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 D10 C4○D8 Dic10 C8⋊C22 D4⋊2D5 D4×D5 D8⋊3D5 D40⋊C2 kernel D4.2Dic10 C20.Q8 C20.8Q8 C40⋊5C4 D4⋊Dic5 C5×D4⋊C4 C4.Dic10 D4×Dic5 C2×Dic5 C5×D4 D4⋊C4 C20 C4⋊C4 C2×C8 C2×D4 C10 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 1 2 2 4 4

Matrix representation of D4.2Dic10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 37 0 0 0 0 31 0 0 0 0 0 0 0 12 12 0 0 0 0 12 29 0 0 0 0 0 0 1 40 0 0 0 0 36 6
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 35 40 0 0 0 0 35 6

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,31,0,0,0,0,37,0,0,0,0,0,0,0,12,12,0,0,0,0,12,29,0,0,0,0,0,0,1,36,0,0,0,0,40,6],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,35,35,0,0,0,0,40,6] >;

D4.2Dic10 in GAP, Magma, Sage, TeX

D_4._2{\rm Dic}_{10}
% in TeX

G:=Group("D4.2Dic10");
// GroupNames label

G:=SmallGroup(320,393);
// by ID

G=gap.SmallGroup(320,393);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,926,219,226,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2*c^10,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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