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G = D20.12D4order 320 = 26·5

12nd non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.12D4, C4.97(D4×D5), Q8⋊C48D5, (C2×D40).3C2, (C2×C8).20D10, D208C45C2, C4⋊C4.157D10, C4.8(C4○D20), C20.129(C2×D4), C52(D4.2D4), (C2×Q8).24D10, C20.8Q88C2, D206C413C2, C10.72(C4○D8), C20.24(C4○D4), C20.23D42C2, (C2×C40).20C22, (C2×Dic5).44D4, C22.208(D4×D5), C2.19(D40⋊C2), C10.28(C4⋊D4), C10.66(C8⋊C22), (C2×C20).259C23, (C2×D20).73C22, (Q8×C10).42C22, C2.11(Q8.D10), C2.31(D10⋊D4), (C4×Dic5).32C22, (C2×Q8⋊D5)⋊6C2, (C5×Q8⋊C4)⋊8C2, (C2×C10).272(C2×D4), (C5×C4⋊C4).60C22, (C2×C52C8).49C22, (C2×C4).366(C22×D5), SmallGroup(320,446)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.12D4
C1C5C10C2×C10C2×C20C2×D20D208C4 — D20.12D4
C5C10C2×C20 — D20.12D4
C1C22C2×C4Q8⋊C4

Generators and relations for D20.12D4
 G = < a,b,c,d | a20=b2=c4=1, d2=a15, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a15b, dcd-1=a5c-1 >

Subgroups: 630 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 [×3], C10 [×3], C42, C22⋊C4 [×3], C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8, C40, C4×D5 [×2], D20 [×2], D20 [×3], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], D4.2D4, D40 [×2], C2×C52C8, C4×Dic5, D10⋊C4 [×3], Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20 [×2], Q8×C10, D206C4, C20.8Q8, C5×Q8⋊C4, D208C4, C2×D40, C2×Q8⋊D5, C20.23D4, D20.12D4
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, D40⋊C2, Q8.D10, D20.12D4

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444455888810···102020202020···2040···40
size111120204022448101020224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C4○D20C8⋊C22D4×D5D4×D5D40⋊C2Q8.D10
kernelD20.12D4D206C4C20.8Q8C5×Q8⋊C4D208C4C2×D40C2×Q8⋊D5C20.23D4D20C2×Dic5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C10C4C10C4C22C2C2
# reps1111111122222224812244

Matrix representation of D20.12D4 in GL4(𝔽41) generated by

04000
1700
00402
00401
,
323000
11900
001724
002924
,
9000
0900
003011
001511
,
303200
91100
002417
00120
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,40,40,0,0,2,1],[32,11,0,0,30,9,0,0,0,0,17,29,0,0,24,24],[9,0,0,0,0,9,0,0,0,0,30,15,0,0,11,11],[30,9,0,0,32,11,0,0,0,0,24,12,0,0,17,0] >;

D20.12D4 in GAP, Magma, Sage, TeX

D_{20}._{12}D_4
% in TeX

G:=Group("D20.12D4");
// GroupNames label

G:=SmallGroup(320,446);
// by ID

G=gap.SmallGroup(320,446);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,184,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^15,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

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