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## G = Dic10.D4order 320 = 26·5

### 8th non-split extension by Dic10 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10.D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — Dic5⋊3Q8 — Dic10.D4
 Lower central C5 — C10 — C2×C20 — Dic10.D4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic10.D4
G = < a,b,c,d | a20=c4=1, b2=d2=a10, bab-1=dad-1=a-1, cac-1=a9, bc=cb, dbd-1=a15b, dcd-1=a10c-1 >

Subgroups: 422 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8, C40, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, Q8.D4, Dic20, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D4.D5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, D4×C10, C10.Q16, C20.8Q8, C5×D4⋊C4, Dic53Q8, C2×Dic20, C2×D4.D5, C20.17D4, Dic10.D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8.C22, C22×D5, Q8.D4, C4○D20, D4×D5, D10⋊D4, D83D5, SD16⋊D5, Dic10.D4

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

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 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 4 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 8 2 2 4 4 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 D4 D5 C4○D4 D10 D10 D10 C4○D8 C4○D20 C8.C22 D4×D5 D4×D5 D8⋊3D5 SD16⋊D5 kernel Dic10.D4 C10.Q16 C20.8Q8 C5×D4⋊C4 Dic5⋊3Q8 C2×Dic20 C2×D4.D5 C20.17D4 Dic10 C2×Dic5 D4⋊C4 C20 C4⋊C4 C2×C8 C2×D4 C10 C4 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 Dic10.D4 in GL4(𝔽41) generated by

 23 0 0 0 33 25 0 0 0 0 40 39 0 0 1 1
,
 10 18 0 0 15 31 0 0 0 0 30 30 0 0 26 11
,
 33 2 0 0 29 8 0 0 0 0 32 0 0 0 0 32
,
 33 2 0 0 30 8 0 0 0 0 32 23 0 0 0 9
G:=sub<GL(4,GF(41))| [23,33,0,0,0,25,0,0,0,0,40,1,0,0,39,1],[10,15,0,0,18,31,0,0,0,0,30,26,0,0,30,11],[33,29,0,0,2,8,0,0,0,0,32,0,0,0,0,32],[33,30,0,0,2,8,0,0,0,0,32,0,0,0,23,9] >;

Dic10.D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,394);
// by ID

G=gap.SmallGroup(320,394);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=1,b^2=d^2=a^10,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^9,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^10*c^-1>;
// generators/relations

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