Copied to
clipboard

G = D20.D4order 320 = 26·5

9th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.9D4, C4.89(D4×D5), D4⋊C414D5, D208C44C2, C4⋊C4.141D10, (C2×D4).32D10, C4.4(C4○D20), C10.Q168C2, C20.113(C2×D4), (C2×C8).118D10, C51(D4.2D4), C10.43(C4○D8), C20.12(C4○D4), C20.17D42C2, C20.8Q812C2, (C2×Dic5).33D4, C22.184(D4×D5), C2.19(D8⋊D5), C10.20(C4⋊D4), C10.37(C8⋊C22), (C2×C20).227C23, (C2×C40).129C22, (C2×D20).59C22, (D4×C10).48C22, C2.23(D10⋊D4), (C4×Dic5).20C22, C2.13(SD163D5), (C2×Dic10).65C22, (C2×D4⋊D5).3C2, (C2×C40⋊C2)⋊16C2, (C5×D4⋊C4)⋊14C2, (C2×C10).240(C2×D4), (C5×C4⋊C4).28C22, (C2×C52C8).25C22, (C2×C4).334(C22×D5), SmallGroup(320,414)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.D4
C1C5C10C2×C10C2×C20C2×D20D208C4 — D20.D4
C5C10C2×C20 — D20.D4
C1C22C2×C4D4⋊C4

Generators and relations for D20.D4
 G = < a,b,c,d | a20=b2=c4=1, d2=a5, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a15b, dcd-1=a5c-1 >

Subgroups: 566 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×5], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C10, C42, C22⋊C4 [×3], C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×4], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8, C40, Dic10 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, D4.2D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, D10⋊C4, D4⋊D5 [×2], C23.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D4×C10, C10.Q16, C20.8Q8, C5×D4⋊C4, D208C4, C2×C40⋊C2, C2×D4⋊D5, C20.17D4, D20.D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, C22×D5, D4.2D4, C4○D20, D4×D5 [×2], D10⋊D4, D8⋊D5, SD163D5, D20.D4

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444444455888810···1010101010202020202020202040···40
size111182020224410102040224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C4○D20C8⋊C22D4×D5D4×D5D8⋊D5SD163D5
kernelD20.D4C10.Q16C20.8Q8C5×D4⋊C4D208C4C2×C40⋊C2C2×D4⋊D5C20.17D4D20C2×Dic5D4⋊C4C20C4⋊C4C2×C8C2×D4C10C4C10C4C22C2C2
# reps1111111122222224812244

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

1900
184000
0065
0011
,
403200
0100
00136
00040
,
94000
03200
0090
0009
,
02900
173000
0024
00939
G:=sub<GL(4,GF(41))| [1,18,0,0,9,40,0,0,0,0,6,1,0,0,5,1],[40,0,0,0,32,1,0,0,0,0,1,0,0,0,36,40],[9,0,0,0,40,32,0,0,0,0,9,0,0,0,0,9],[0,17,0,0,29,30,0,0,0,0,2,9,0,0,4,39] >;

D20.D4 in GAP, Magma, Sage, TeX

D_{20}.D_4
% in TeX

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

G:=SmallGroup(320,414);
// by ID

G=gap.SmallGroup(320,414);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

׿
×
𝔽