Copied to
clipboard

G = D2017D4order 320 = 26·5

5th semidirect product of D20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2017D4, Dic1016D4, C4⋊D42D5, C4.99(D4×D5), C54(D4⋊D4), C4⋊C4.57D10, (C2×D4).37D10, C20.146(C2×D4), (C2×C20).262D4, D206C434C2, C10.45C22≀C2, C10.96(C4○D8), C10.Q1633C2, (C22×C10).83D4, C20.55D411C2, C10.90(C8⋊C22), (C2×C20).356C23, (D4×C10).53C22, (C22×C4).120D10, C23.23(C5⋊D4), C2.13(C23⋊D10), (C2×D20).249C22, C2.15(D4.8D10), C2.11(D4.D10), (C22×C20).160C22, (C2×Dic10).276C22, (C2×D4⋊D5)⋊9C2, (C5×C4⋊D4)⋊2C2, (C2×D4.D5)⋊8C2, (C2×C4○D20)⋊15C2, (C2×C10).487(C2×D4), (C2×C4).171(C5⋊D4), (C5×C4⋊C4).104C22, (C2×C4).456(C22×D5), C22.162(C2×C5⋊D4), (C2×C52C8).108C22, SmallGroup(320,664)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D2017D4
C1C5C10C20C2×C20C2×D20C2×C4○D20 — D2017D4
C5C10C2×C20 — D2017D4
C1C22C22×C4C4⋊D4

Generators and relations for D2017D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a5b, dbd=a10b, dcd=c-1 >

Subgroups: 670 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8 [×3], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C22×C4, C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×6], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×D5, C22×C10, C22×C10, D4⋊D4, C2×C52C8 [×2], D4⋊D5 [×2], D4.D5 [×2], C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C2×C5⋊D4, C22×C20, D4×C10, D4×C10, D206C4, C10.Q16, C20.55D4, C2×D4⋊D5, C2×D4.D5, C5×C4⋊D4, C2×C4○D20, D2017D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4⋊D4, D4×D5 [×2], C2×C5⋊D4, D4.D10, C23⋊D10, D4.8D10, D2017D4

Smallest permutation representation of D2017D4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(121 131)(122 130)(123 129)(124 128)(125 127)(132 140)(133 139)(134 138)(135 137)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 160)
(1 139 67 142)(2 130 68 153)(3 121 69 144)(4 132 70 155)(5 123 71 146)(6 134 72 157)(7 125 73 148)(8 136 74 159)(9 127 75 150)(10 138 76 141)(11 129 77 152)(12 140 78 143)(13 131 79 154)(14 122 80 145)(15 133 61 156)(16 124 62 147)(17 135 63 158)(18 126 64 149)(19 137 65 160)(20 128 66 151)(21 53 108 93)(22 44 109 84)(23 55 110 95)(24 46 111 86)(25 57 112 97)(26 48 113 88)(27 59 114 99)(28 50 115 90)(29 41 116 81)(30 52 117 92)(31 43 118 83)(32 54 119 94)(33 45 120 85)(34 56 101 96)(35 47 102 87)(36 58 103 98)(37 49 104 89)(38 60 105 100)(39 51 106 91)(40 42 107 82)
(1 40)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(41 141)(42 142)(43 143)(44 144)(45 145)(46 146)(47 147)(48 148)(49 149)(50 150)(51 151)(52 152)(53 153)(54 154)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 138)(82 139)(83 140)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222222444444455888810···10101010101010101020···2020202020
size111148202022228202022202020202···2444488884···48888

47 irreducible representations

dim11111111222222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8C5⋊D4C5⋊D4C8⋊C22D4×D5D4.D10D4.8D10
kernelD2017D4D206C4C10.Q16C20.55D4C2×D4⋊D5C2×D4.D5C5×C4⋊D4C2×C4○D20Dic10D20C2×C20C22×C10C4⋊D4C4⋊C4C22×C4C2×D4C10C2×C4C23C10C4C2C2
# reps11111111221122224441444

Matrix representation of D2017D4 in GL6(𝔽41)

4000000
0400000
0035100
0054000
0000121
00003740
,
4000000
3910000
00404000
000100
0000121
0000040
,
9320000
0320000
0040000
0004000
0000013
0000220
,
100000
2400000
001000
000100
00003216
0000369

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[9,0,0,0,0,0,32,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,22,0,0,0,0,13,0],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,36,0,0,0,0,16,9] >;

D2017D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{17}D_4
% in TeX

G:=Group("D20:17D4");
// GroupNames label

G:=SmallGroup(320,664);
// by ID

G=gap.SmallGroup(320,664);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,1123,297,136,12550]);
// Polycyclic

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

׿
×
𝔽