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G = Dic1017D4order 320 = 26·5

5th semidirect product of Dic10 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1017D4, C4⋊C4.60D10, C53(Q8⋊D4), (C2×C10)⋊3SD16, C4⋊D4.5D5, C4.100(D4×D5), (C2×C20).73D4, (C2×D4).40D10, C20.149(C2×D4), C10.46C22≀C2, C10.Q1634C2, C222(D4.D5), C10.55(C2×SD16), (C22×C10).86D4, C20.55D412C2, (C2×C20).359C23, (D4×C10).56C22, (C22×C4).122D10, C23.59(C5⋊D4), C2.14(C23⋊D10), (C22×Dic10)⋊13C2, C2.12(D4.9D10), C10.114(C8.C22), (C22×C20).163C22, (C2×Dic10).277C22, (C2×D4.D5)⋊9C2, C2.9(C2×D4.D5), (C5×C4⋊D4).4C2, (C2×C10).490(C2×D4), (C2×C4).51(C5⋊D4), (C5×C4⋊C4).107C22, (C2×C4).459(C22×D5), C22.165(C2×C5⋊D4), (C2×C52C8).110C22, SmallGroup(320,667)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic1017D4
C1C5C10C20C2×C20C2×Dic10C22×Dic10 — Dic1017D4
C5C10C2×C20 — Dic1017D4
C1C22C22×C4C4⋊D4

Generators and relations for Dic1017D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >

Subgroups: 574 in 158 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8 [×10], C23, C23, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8 [×7], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×5], C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, C52C8 [×2], Dic10 [×4], Dic10 [×6], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×C10, C22×C10, Q8⋊D4, C2×C52C8 [×2], D4.D5 [×4], C5×C22⋊C4, C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×5], C22×Dic5, C22×C20, D4×C10, D4×C10, C10.Q16 [×2], C20.55D4, C2×D4.D5 [×2], C5×C4⋊D4, C22×Dic10, Dic1017D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8.C22, C5⋊D4 [×2], C22×D5, Q8⋊D4, D4.D5 [×2], D4×D5 [×2], C2×C5⋊D4, C2×D4.D5, C23⋊D10, D4.9D10, Dic1017D4

Smallest permutation representation of Dic1017D4
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 61 11 71)(2 80 12 70)(3 79 13 69)(4 78 14 68)(5 77 15 67)(6 76 16 66)(7 75 17 65)(8 74 18 64)(9 73 19 63)(10 72 20 62)(21 152 31 142)(22 151 32 141)(23 150 33 160)(24 149 34 159)(25 148 35 158)(26 147 36 157)(27 146 37 156)(28 145 38 155)(29 144 39 154)(30 143 40 153)(41 138 51 128)(42 137 52 127)(43 136 53 126)(44 135 54 125)(45 134 55 124)(46 133 56 123)(47 132 57 122)(48 131 58 121)(49 130 59 140)(50 129 60 139)(81 105 91 115)(82 104 92 114)(83 103 93 113)(84 102 94 112)(85 101 95 111)(86 120 96 110)(87 119 97 109)(88 118 98 108)(89 117 99 107)(90 116 100 106)
(1 131 153 98)(2 122 154 89)(3 133 155 100)(4 124 156 91)(5 135 157 82)(6 126 158 93)(7 137 159 84)(8 128 160 95)(9 139 141 86)(10 130 142 97)(11 121 143 88)(12 132 144 99)(13 123 145 90)(14 134 146 81)(15 125 147 92)(16 136 148 83)(17 127 149 94)(18 138 150 85)(19 129 151 96)(20 140 152 87)(21 104 72 54)(22 115 73 45)(23 106 74 56)(24 117 75 47)(25 108 76 58)(26 119 77 49)(27 110 78 60)(28 101 79 51)(29 112 80 42)(30 103 61 53)(31 114 62 44)(32 105 63 55)(33 116 64 46)(34 107 65 57)(35 118 66 48)(36 109 67 59)(37 120 68 50)(38 111 69 41)(39 102 70 52)(40 113 71 43)
(1 98)(2 99)(3 100)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(61 108)(62 109)(63 110)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 120)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(121 143)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 153)(132 154)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 141)(140 142)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222224444444455888810···10101010101010101020···2020202020
size111122822482020202022202020202···2444488884···48888

47 irreducible representations

dim11111122222222224444
type+++++++++++++-+--
imageC1C2C2C2C2C2D4D4D4D5SD16D10D10D10C5⋊D4C5⋊D4C8.C22D4×D5D4.D5D4.9D10
kernelDic1017D4C10.Q16C20.55D4C2×D4.D5C5×C4⋊D4C22×Dic10Dic10C2×C20C22×C10C4⋊D4C2×C10C4⋊C4C22×C4C2×D4C2×C4C23C10C4C22C2
# reps12121141124222441444

Matrix representation of Dic1017D4 in GL6(𝔽41)

40320000
2310000
0040100
0053500
000010
000001
,
11290000
17300000
0038600
0026300
000010
000001
,
100000
18400000
001000
000100
00004039
000011
,
100000
010000
001000
000100
00004039
000001

G:=sub<GL(6,GF(41))| [40,23,0,0,0,0,32,1,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,17,0,0,0,0,29,30,0,0,0,0,0,0,38,26,0,0,0,0,6,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,18,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1] >;

Dic1017D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{17}D_4
% in TeX

G:=Group("Dic10:17D4");
// GroupNames label

G:=SmallGroup(320,667);
// by ID

G=gap.SmallGroup(320,667);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
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