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G = D4.1D20order 320 = 26·5

1st non-split extension by D4 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.1D20, C42.50D10, (C4×D4)⋊4D5, (D4×C20)⋊4C2, C203C824C2, (C5×D4).18D4, C20.18(C2×D4), C4.14(C2×D20), (C2×C20).61D4, C4⋊C4.244D10, C54(D4.2D4), D206C431C2, (C2×D4).191D10, C4.10(C4○D20), C10.89(C4○D8), C20.51(C4○D4), C4.D2013C2, C10.Q1629C2, (C4×C20).87C22, C2.13(C207D4), C10.65(C4⋊D4), C10.87(C8⋊C22), (C2×C20).338C23, (C2×D20).98C22, C2.9(D4.D10), (D4×C10).233C22, C2.11(D4.8D10), (C2×Dic10).103C22, (C2×D4⋊D5).5C2, (C2×D4.D5)⋊7C2, (C2×C10).469(C2×D4), (C2×C4).218(C5⋊D4), (C5×C4⋊C4).275C22, (C2×C52C8).94C22, (C2×C4).438(C22×D5), C22.150(C2×C5⋊D4), SmallGroup(320,643)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.1D20
C1C5C10C20C2×C20C2×D20C4.D20 — D4.1D20
C5C10C2×C20 — D4.1D20
C1C22C42C4×D4

Generators and relations for D4.1D20
 G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 502 in 124 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], D4 [×2], D4 [×3], Q8 [×2], C23 [×2], D5, C10 [×3], C10 [×2], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×4], D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8 [×2], Dic10 [×2], D20 [×2], C2×Dic5, C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×D5, C22×C10, D4.2D4, C2×C52C8 [×2], D10⋊C4 [×2], D4⋊D5 [×2], D4.D5 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, C22×C20, D4×C10, C203C8, D206C4, C10.Q16, C4.D20, C2×D4⋊D5, C2×D4.D5, D4×C20, D4.1D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, D20 [×2], C5⋊D4 [×2], C22×D5, D4.2D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.D10, D4.8D10, D4.1D20

Smallest permutation representation of D4.1D20
On 160 points
Generators in S160
(1 80 113 145)(2 61 114 146)(3 62 115 147)(4 63 116 148)(5 64 117 149)(6 65 118 150)(7 66 119 151)(8 67 120 152)(9 68 101 153)(10 69 102 154)(11 70 103 155)(12 71 104 156)(13 72 105 157)(14 73 106 158)(15 74 107 159)(16 75 108 160)(17 76 109 141)(18 77 110 142)(19 78 111 143)(20 79 112 144)(21 133 85 42)(22 134 86 43)(23 135 87 44)(24 136 88 45)(25 137 89 46)(26 138 90 47)(27 139 91 48)(28 140 92 49)(29 121 93 50)(30 122 94 51)(31 123 95 52)(32 124 96 53)(33 125 97 54)(34 126 98 55)(35 127 99 56)(36 128 100 57)(37 129 81 58)(38 130 82 59)(39 131 83 60)(40 132 84 41)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(61 98)(62 99)(63 100)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)
(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)
(1 112 113 20)(2 19 114 111)(3 110 115 18)(4 17 116 109)(5 108 117 16)(6 15 118 107)(7 106 119 14)(8 13 120 105)(9 104 101 12)(10 11 102 103)(21 45 85 136)(22 135 86 44)(23 43 87 134)(24 133 88 42)(25 41 89 132)(26 131 90 60)(27 59 91 130)(28 129 92 58)(29 57 93 128)(30 127 94 56)(31 55 95 126)(32 125 96 54)(33 53 97 124)(34 123 98 52)(35 51 99 122)(36 121 100 50)(37 49 81 140)(38 139 82 48)(39 47 83 138)(40 137 84 46)(61 143 146 78)(62 77 147 142)(63 141 148 76)(64 75 149 160)(65 159 150 74)(66 73 151 158)(67 157 152 72)(68 71 153 156)(69 155 154 70)(79 145 144 80)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order12222224444444455888810···1010···1020···2020···20
size1111444022224444022202020202···24···42···24···4

59 irreducible representations

dim1111111122222222222444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C5⋊D4D20C4○D20C8⋊C22D4.D10D4.8D10
kernelD4.1D20C203C8D206C4C10.Q16C4.D20C2×D4⋊D5C2×D4.D5D4×C20C2×C20C5×D4C4×D4C20C42C4⋊C4C2×D4C10C2×C4D4C4C10C2C2
# reps1111111122222224888144

Matrix representation of D4.1D20 in GL4(𝔽41) generated by

1000
0100
00121
003740
,
40000
04000
00035
00340
,
163000
27200
00320
00032
,
393000
4200
00320
00369
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,37,0,0,21,40],[40,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0],[16,27,0,0,30,2,0,0,0,0,32,0,0,0,0,32],[39,4,0,0,30,2,0,0,0,0,32,36,0,0,0,9] >;

D4.1D20 in GAP, Magma, Sage, TeX

D_4._1D_{20}
% in TeX

G:=Group("D4.1D20");
// GroupNames label

G:=SmallGroup(320,643);
// by ID

G=gap.SmallGroup(320,643);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1123,297,136,12550]);
// Polycyclic

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

׿
×
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