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G = Dic102D4order 320 = 26·5

2nd semidirect product of Dic10 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic102D4, Dic54SD16, C4.82(D4×D5), D4⋊C49D5, C51(C4⋊SD16), D206C42C2, C4⋊C4.131D10, (C2×D4).18D10, C20.2(C4○D4), C4.1(C4○D20), C20⋊D4.5C2, (C2×C8).113D10, C20.101(C2×D4), C2.10(D5×SD16), Dic53Q83C2, C2.9(D8⋊D5), C20.8Q810C2, C10.21(C2×SD16), C22.164(D4×D5), C10.13(C4⋊D4), C10.26(C8⋊C22), (C2×C40).124C22, (C2×C20).202C23, (C2×Dic5).192D4, (C2×D20).49C22, (D4×C10).23C22, C2.16(D10⋊D4), (C4×Dic5).12C22, (C2×Dic10).55C22, (C2×D4.D5)⋊2C2, (C5×D4⋊C4)⋊9C2, (C2×C40⋊C2)⋊13C2, (C5×C4⋊C4).7C22, (C2×C10).215(C2×D4), (C2×C52C8).8C22, (C2×C4).309(C22×D5), SmallGroup(320,389)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic102D4
C1C5C10C20C2×C20C4×Dic5C20⋊D4 — Dic102D4
C5C10C2×C20 — Dic102D4
C1C22C2×C4D4⋊C4

Generators and relations for Dic102D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, bc=cb, dbd=a15b, dcd=c-1 >

Subgroups: 614 in 128 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×4], D4 [×8], Q8 [×3], C23 [×2], D5, C10 [×3], C10, C42 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×4], C2×D4, C2×D4 [×3], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], D4⋊C4, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C52C8, C40, Dic10 [×2], Dic10, D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C4⋊SD16, C40⋊C2 [×2], C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D4.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, C2×C5⋊D4 [×2], D4×C10, D206C4, C20.8Q8, C5×D4⋊C4, Dic53Q8, C2×C40⋊C2, C2×D4.D5, C20⋊D4, Dic102D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8⋊C22, C22×D5, C4⋊SD16, C4○D20, D4×D5 [×2], D10⋊D4, D8⋊D5, D5×SD16, Dic102D4

Smallest permutation representation of Dic102D4
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 95 11 85)(2 94 12 84)(3 93 13 83)(4 92 14 82)(5 91 15 81)(6 90 16 100)(7 89 17 99)(8 88 18 98)(9 87 19 97)(10 86 20 96)(21 153 31 143)(22 152 32 142)(23 151 33 141)(24 150 34 160)(25 149 35 159)(26 148 36 158)(27 147 37 157)(28 146 38 156)(29 145 39 155)(30 144 40 154)(41 113 51 103)(42 112 52 102)(43 111 53 101)(44 110 54 120)(45 109 55 119)(46 108 56 118)(47 107 57 117)(48 106 58 116)(49 105 59 115)(50 104 60 114)(61 139 71 129)(62 138 72 128)(63 137 73 127)(64 136 74 126)(65 135 75 125)(66 134 76 124)(67 133 77 123)(68 132 78 122)(69 131 79 121)(70 130 80 140)
(1 29 133 51)(2 38 134 60)(3 27 135 49)(4 36 136 58)(5 25 137 47)(6 34 138 56)(7 23 139 45)(8 32 140 54)(9 21 121 43)(10 30 122 52)(11 39 123 41)(12 28 124 50)(13 37 125 59)(14 26 126 48)(15 35 127 57)(16 24 128 46)(17 33 129 55)(18 22 130 44)(19 31 131 53)(20 40 132 42)(61 119 99 141)(62 108 100 150)(63 117 81 159)(64 106 82 148)(65 115 83 157)(66 104 84 146)(67 113 85 155)(68 102 86 144)(69 111 87 153)(70 120 88 142)(71 109 89 151)(72 118 90 160)(73 107 91 149)(74 116 92 158)(75 105 93 147)(76 114 94 156)(77 103 95 145)(78 112 96 154)(79 101 97 143)(80 110 98 152)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 60)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(95 100)(96 99)(97 98)(101 152)(102 151)(103 150)(104 149)(105 148)(106 147)(107 146)(108 145)(109 144)(110 143)(111 142)(112 141)(113 160)(114 159)(115 158)(116 157)(117 156)(118 155)(119 154)(120 153)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444455888810···1010101010202020202020202040···40
size111184022441010202020224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5SD16C4○D4D10D10D10C4○D20C8⋊C22D4×D5D4×D5D8⋊D5D5×SD16
kernelDic102D4D206C4C20.8Q8C5×D4⋊C4Dic53Q8C2×C40⋊C2C2×D4.D5C20⋊D4Dic10C2×Dic5D4⋊C4Dic5C20C4⋊C4C2×C8C2×D4C4C10C4C22C2C2
# reps1111111122242222812244

Matrix representation of Dic102D4 in GL4(𝔽41) generated by

354000
1000
004039
0011
,
252500
391600
00011
00260
,
212100
182000
00400
00040
,
63500
403500
0010
004040
G:=sub<GL(4,GF(41))| [35,1,0,0,40,0,0,0,0,0,40,1,0,0,39,1],[25,39,0,0,25,16,0,0,0,0,0,26,0,0,11,0],[21,18,0,0,21,20,0,0,0,0,40,0,0,0,0,40],[6,40,0,0,35,35,0,0,0,0,1,40,0,0,0,40] >;

Dic102D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_2D_4
% in TeX

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

G:=SmallGroup(320,389);
// by ID

G=gap.SmallGroup(320,389);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,135,268,570,297,136,12550]);
// 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,c*a*c^-1=a^9,b*c=c*b,d*b*d=a^15*b,d*c*d=c^-1>;
// generators/relations

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