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G = Dic1022D4order 320 = 26·5

10th semidirect product of Dic10 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1022D4, C10.192- 1+4, C54(Q85D4), C4.114(D4×D5), C22⋊Q810D5, C4⋊D2026C2, C4⋊C4.191D10, C20.237(C2×D4), D1014(C4○D4), D10⋊D426C2, D208C427C2, D10⋊Q820C2, (C2×C20).56C23, (C2×Q8).128D10, C22⋊C4.17D10, Dic5.49(C2×D4), C10.79(C22×D4), Dic53Q826C2, C20.23D413C2, (C2×C10).177C24, (C22×C4).239D10, Dic5.5D425C2, (C2×D20).273C22, (Q8×C10).109C22, C22.198(C23×D5), C23.120(C22×D5), (C22×C10).205C23, (C22×C20).257C22, (C2×Dic5).246C23, (C4×Dic5).115C22, C10.D4.29C22, (C22×D5).209C23, C23.D5.118C22, D10⋊C4.128C22, C2.20(Q8.10D10), (C2×Dic10).303C22, (C2×Q8×D5)⋊8C2, C2.52(C2×D4×D5), (C4×C5⋊D4)⋊24C2, C2.50(D5×C4○D4), (C2×C4○D20)⋊25C2, (C5×C22⋊Q8)⋊13C2, C10.162(C2×C4○D4), (C2×C4×D5).106C22, (C5×C4⋊C4).160C22, (C2×C4).592(C22×D5), (C2×C5⋊D4).133C22, (C5×C22⋊C4).32C22, SmallGroup(320,1305)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic1022D4
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q8×D5 — Dic1022D4
C5C2×C10 — Dic1022D4
C1C22C22⋊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 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×13], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], D4 [×12], Q8 [×10], C23, C23 [×3], D5 [×4], C10 [×3], C10, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], Dic5 [×3], C20 [×2], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C10 [×3], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8, C22⋊Q8 [×2], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×4], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×C10, Q85D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], D10⋊C4, D10⋊C4 [×6], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], C4○D20 [×4], Q8×D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, Q8×C10, D10⋊D4 [×2], Dic5.5D4 [×2], Dic53Q8, D208C4 [×2], C4⋊D20, D10⋊Q8 [×2], C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C2×C4○D20, C2×Q8×D5, Dic1022D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C22×D5 [×7], Q85D4, D4×D5 [×2], 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)
(1 121 11 131)(2 140 12 130)(3 139 13 129)(4 138 14 128)(5 137 15 127)(6 136 16 126)(7 135 17 125)(8 134 18 124)(9 133 19 123)(10 132 20 122)(21 105 31 115)(22 104 32 114)(23 103 33 113)(24 102 34 112)(25 101 35 111)(26 120 36 110)(27 119 37 109)(28 118 38 108)(29 117 39 107)(30 116 40 106)(41 72 51 62)(42 71 52 61)(43 70 53 80)(44 69 54 79)(45 68 55 78)(46 67 56 77)(47 66 57 76)(48 65 58 75)(49 64 59 74)(50 63 60 73)(81 142 91 152)(82 141 92 151)(83 160 93 150)(84 159 94 149)(85 158 95 148)(86 157 96 147)(87 156 97 146)(88 155 98 145)(89 154 99 144)(90 153 100 143)
(1 28 69 98)(2 37 70 87)(3 26 71 96)(4 35 72 85)(5 24 73 94)(6 33 74 83)(7 22 75 92)(8 31 76 81)(9 40 77 90)(10 29 78 99)(11 38 79 88)(12 27 80 97)(13 36 61 86)(14 25 62 95)(15 34 63 84)(16 23 64 93)(17 32 65 82)(18 21 66 91)(19 30 67 100)(20 39 68 89)(41 158 128 111)(42 147 129 120)(43 156 130 109)(44 145 131 118)(45 154 132 107)(46 143 133 116)(47 152 134 105)(48 141 135 114)(49 150 136 103)(50 159 137 112)(51 148 138 101)(52 157 139 110)(53 146 140 119)(54 155 121 108)(55 144 122 117)(56 153 123 106)(57 142 124 115)(58 151 125 104)(59 160 126 113)(60 149 127 102)
(1 155)(2 144)(3 153)(4 142)(5 151)(6 160)(7 149)(8 158)(9 147)(10 156)(11 145)(12 154)(13 143)(14 152)(15 141)(16 150)(17 159)(18 148)(19 157)(20 146)(21 51)(22 60)(23 49)(24 58)(25 47)(26 56)(27 45)(28 54)(29 43)(30 52)(31 41)(32 50)(33 59)(34 48)(35 57)(36 46)(37 55)(38 44)(39 53)(40 42)(61 116)(62 105)(63 114)(64 103)(65 112)(66 101)(67 110)(68 119)(69 108)(70 117)(71 106)(72 115)(73 104)(74 113)(75 102)(76 111)(77 120)(78 109)(79 118)(80 107)(81 128)(82 137)(83 126)(84 135)(85 124)(86 133)(87 122)(88 131)(89 140)(90 129)(91 138)(92 127)(93 136)(94 125)(95 134)(96 123)(97 132)(98 121)(99 130)(100 139)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I···4N4O4P5A5B10A···10F10G10H10I10J20A···20H20I···20P
order122222222444444444···4445510···101010101020···2020···20
size11114101020202222444410···102020222···244444···48···8

53 irreducible representations

dim11111111111122222224444
type++++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D102- 1+4D4×D5Q8.10D10D5×C4○D4
kernelDic1022D4D10⋊D4Dic5.5D4Dic53Q8D208C4C4⋊D20D10⋊Q8C4×C5⋊D4C20.23D4C5×C22⋊Q8C2×C4○D20C2×Q8×D5Dic10C22⋊Q8D10C22⋊C4C4⋊C4C22×C4C2×Q8C10C4C2C2
# reps12212121111142446221444

Matrix representation of Dic1022D4 in GL6(𝔽41)

090000
900000
006100
0040000
0000400
0000040
,
0400000
100000
0035600
001600
0000400
0000040
,
010000
100000
0063500
00403500
0000123
00003240
,
100000
010000
0035600
001600
00004018
000001

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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