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G = D20.7D4order 320 = 26·5

7th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.7D4, Dic10.7D4, M4(2).6D10, C4.153(D4×D5), C52C8.43D4, (C2×Q8).8D10, C4.10D44D5, C20.100(C2×D4), C51(D4.5D4), C20.53D44C2, C4.12D208C2, (C2×C20).12C23, C8.D10.1C2, C4○D20.8C22, C10.12(C4⋊D4), D20.2C4.2C2, C20.C23.1C2, (Q8×C10).10C22, C2.15(D10⋊D4), C4.Dic5.7C22, C22.16(C4○D20), (C2×Dic10).52C22, (C5×M4(2)).15C22, (C2×C5⋊Q16)⋊1C2, (C5×C4.10D4)⋊2C2, (C2×C52C8).4C22, (C2×C4).12(C22×D5), (C2×C10).33(C4○D4), SmallGroup(320,382)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.7D4
C1C5C10C20C2×C20C4○D20D20.2C4 — D20.7D4
C5C10C2×C20 — D20.7D4
C1C2C2×C4C4.10D4

Generators and relations for D20.7D4
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a5c3 >

Subgroups: 382 in 100 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×5], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], D5, C10, C10, C2×C8 [×2], M4(2) [×2], M4(2) [×2], SD16 [×2], Q16 [×4], C2×Q8, C2×Q8, C4○D4, Dic5 [×2], C20 [×2], C20, D10, C2×C10, C4.10D4, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22 [×2], C52C8 [×2], C52C8, C40 [×2], Dic10, Dic10 [×2], C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8 [×2], D4.5D4, C8×D5, C8⋊D5, C40⋊C2, Dic20, C2×C52C8, C4.Dic5, Q8⋊D5, C5⋊Q16 [×3], C5×M4(2) [×2], C2×Dic10, C4○D20, Q8×C10, C20.53D4, C4.12D20, C5×C4.10D4, D20.2C4, C8.D10, C20.C23, C2×C5⋊Q16, D20.7D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C22×D5, D4.5D4, C4○D20, D4×D5 [×2], D10⋊D4, D20.7D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C8D8E8F8G10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12224444455888888810101010202020202020202040···40
size11220228204022448101020402244444488888···8

38 irreducible representations

dim1111111122222222448
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10C4○D20D4.5D4D4×D5D20.7D4
kernelD20.7D4C20.53D4C4.12D20C5×C4.10D4D20.2C4C8.D10C20.C23C2×C5⋊Q16C52C8Dic10D20C4.10D4C2×C10M4(2)C2×Q8C22C5C4C1
# reps1111111121122428242

Matrix representation of D20.7D4 in GL8(𝔽41)

040000000
17000000
000400000
00170000
00004018039
0000040932
00001001
0000123402
,
040000000
400000000
00010000
00100000
00004018039
000032403232
00001001
000012312
,
400000000
040000000
00100000
00010000
000000170
00000171515
0000120240
00002930240
,
00100000
00010000
400000000
040000000
00003336433
00001042323
000061026
00001036392

G:=sub<GL(8,GF(41))| [0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,40,0,1,1,0,0,0,0,18,40,0,23,0,0,0,0,0,9,0,40,0,0,0,0,39,32,1,2],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,32,1,1,0,0,0,0,18,40,0,23,0,0,0,0,0,32,0,1,0,0,0,0,39,32,1,2],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,29,0,0,0,0,0,17,0,30,0,0,0,0,17,15,24,24,0,0,0,0,0,15,0,0],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,33,10,6,10,0,0,0,0,36,4,10,36,0,0,0,0,4,23,2,39,0,0,0,0,33,23,6,2] >;

D20.7D4 in GAP, Magma, Sage, TeX

D_{20}._7D_4
% in TeX

G:=Group("D20.7D4");
// GroupNames label

G:=SmallGroup(320,382);
// by ID

G=gap.SmallGroup(320,382);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,184,297,136,1684,851,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^3>;
// generators/relations

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