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## G = Dic10⋊8D4order 320 = 26·5

### 1st semidirect product of Dic10 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10⋊8D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C4×Dic10 — Dic10⋊8D4
 Lower central C5 — C10 — C2×C20 — Dic10⋊8D4
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for Dic108D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a5b, dcd=c-1 >

Subgroups: 710 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16, C40, Dic10, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C4⋊SD16, C40⋊C2, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C2×D20, C2×D20, D205C4, C5×C4⋊C8, C4×Dic10, C204D4, C2×C40⋊C2, Dic108D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8⋊C22, D20, C22×D5, C4⋊SD16, C40⋊C2, C2×D20, D4×D5, Q82D5, C4⋊D20, C2×C40⋊C2, C8⋊D10, Dic108D4

Smallest permutation representation of Dic108D4
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 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 74 31 64)(22 73 32 63)(23 72 33 62)(24 71 34 61)(25 70 35 80)(26 69 36 79)(27 68 37 78)(28 67 38 77)(29 66 39 76)(30 65 40 75)(81 132 91 122)(82 131 92 121)(83 130 93 140)(84 129 94 139)(85 128 95 138)(86 127 96 137)(87 126 97 136)(88 125 98 135)(89 124 99 134)(90 123 100 133)(101 151 111 141)(102 150 112 160)(103 149 113 159)(104 148 114 158)(105 147 115 157)(106 146 116 156)(107 145 117 155)(108 144 118 154)(109 143 119 153)(110 142 120 152)
(1 111 74 99)(2 112 75 100)(3 113 76 81)(4 114 77 82)(5 115 78 83)(6 116 79 84)(7 117 80 85)(8 118 61 86)(9 119 62 87)(10 120 63 88)(11 101 64 89)(12 102 65 90)(13 103 66 91)(14 104 67 92)(15 105 68 93)(16 106 69 94)(17 107 70 95)(18 108 71 96)(19 109 72 97)(20 110 73 98)(21 124 60 151)(22 125 41 152)(23 126 42 153)(24 127 43 154)(25 128 44 155)(26 129 45 156)(27 130 46 157)(28 131 47 158)(29 132 48 159)(30 133 49 160)(31 134 50 141)(32 135 51 142)(33 136 52 143)(34 137 53 144)(35 138 54 145)(36 139 55 146)(37 140 56 147)(38 121 57 148)(39 122 58 149)(40 123 59 150)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 120)(13 119)(14 118)(15 117)(16 116)(17 115)(18 114)(19 113)(20 112)(21 139)(22 138)(23 137)(24 136)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 140)(41 145)(42 144)(43 143)(44 142)(45 141)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)

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

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

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

59 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 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 40 40 2 2 2 2 4 20 20 20 20 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 SD16 C4○D4 D10 D10 D20 C40⋊C2 C8⋊C22 D4×D5 Q8⋊2D5 C8⋊D10 kernel Dic10⋊8D4 D20⋊5C4 C5×C4⋊C8 C4×Dic10 C20⋊4D4 C2×C40⋊C2 Dic10 C2×C20 C4⋊C8 C20 C20 C42 C2×C8 C2×C4 C4 C10 C4 C4 C2 # reps 1 2 1 1 1 2 2 2 2 4 2 2 4 8 16 1 2 2 4

Matrix representation of Dic108D4 in GL6(𝔽41)

 1 9 0 0 0 0 18 40 0 0 0 0 0 0 0 40 0 0 0 0 1 35 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 12 0 0 0 0 17 0 0 0 0 0 0 0 2 13 0 0 0 0 25 39 0 0 0 0 0 0 1 4 0 0 0 0 0 40
,
 40 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 32 5 0 0 0 0 0 9
,
 40 0 0 0 0 0 23 1 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 0 0 0 0 9 36 0 0 0 0 16 32

G:=sub<GL(6,GF(41))| [1,18,0,0,0,0,9,40,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,2,25,0,0,0,0,13,39,0,0,0,0,0,0,1,0,0,0,0,0,4,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,5,9],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,36,32] >;

Dic108D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_8D_4
% in TeX

G:=Group("Dic10:8D4");
// GroupNames label

G:=SmallGroup(320,475);
// by ID

G=gap.SmallGroup(320,475);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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