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G = D20.17D4order 320 = 26·5

17th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.17D4, (C2×Q16)⋊4D5, C4.68(D4×D5), C20.53(C2×D4), (C2×C8).39D10, (C5×Q8).10D4, D103Q87C2, (C10×Q16)⋊14C2, C56(D4.7D4), (C2×Q8).63D10, Q8.9(C5⋊D4), D205C430C2, D101C829C2, C10.62C22≀C2, C10.80(C4○D8), Q8⋊Dic534C2, (C22×D5).48D4, C22.278(D4×D5), (C2×C20).461C23, (C2×C40).253C22, (C2×Dic5).244D4, (Q8×C10).90C22, C2.30(C23⋊D10), C2.17(Q8.D10), C2.29(Q16⋊D5), (C2×D20).129C22, C10.79(C8.C22), C4⋊Dic5.184C22, (C2×Q8⋊D5)⋊20C2, C4.49(C2×C5⋊D4), (C2×C4×D5).57C22, (C2×C10).372(C2×D4), (C2×Q82D5).5C2, (C2×C4).549(C22×D5), (C2×C52C8).166C22, SmallGroup(320,814)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.17D4
C1C5C10C20C2×C20C2×C4×D5D103Q8 — D20.17D4
C5C10C2×C20 — D20.17D4
C1C22C2×C4C2×Q16

Generators and relations for D20.17D4
 G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=cac-1=a-1, dad-1=a11, cbc-1=a3b, dbd-1=a5b, dcd-1=a10c-1 >

Subgroups: 670 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×10], D4 [×7], Q8 [×2], Q8 [×3], C23 [×2], D5 [×3], C10 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×2], Q16 [×2], C22×C4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×3], D10 [×7], C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8, C40, C4×D5 [×6], D20 [×2], D20 [×5], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8 [×3], C22×D5, C22×D5, D4.7D4, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5 [×2], C2×C40, C5×Q16 [×2], C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5 [×4], Q8×C10 [×2], D101C8, D205C4, Q8⋊Dic5, C2×Q8⋊D5, D103Q8, C10×Q16, C2×Q82D5, D20.17D4
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, D4.7D4, D4×D5 [×2], C2×C5⋊D4, Q16⋊D5, Q8.D10, C23⋊D10, D20.17D4

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444455888810···102020202020···2040···40
size111120202022448101040224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++++-+++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10C4○D8C5⋊D4C8.C22D4×D5D4×D5Q16⋊D5Q8.D10
kernelD20.17D4D101C8D205C4Q8⋊Dic5C2×Q8⋊D5D103Q8C10×Q16C2×Q82D5D20C2×Dic5C5×Q8C22×D5C2×Q16C2×C8C2×Q8C10Q8C10C4C22C2C2
# reps1111111121212244812244

Matrix representation of D20.17D4 in GL4(𝔽41) generated by

74000
1000
0011
003940
,
40700
0100
00400
0021
,
3300
243800
00012
00240
,
17100
402400
001126
003030
G:=sub<GL(4,GF(41))| [7,1,0,0,40,0,0,0,0,0,1,39,0,0,1,40],[40,0,0,0,7,1,0,0,0,0,40,2,0,0,0,1],[3,24,0,0,3,38,0,0,0,0,0,24,0,0,12,0],[17,40,0,0,1,24,0,0,0,0,11,30,0,0,26,30] >;

D20.17D4 in GAP, Magma, Sage, TeX

D_{20}._{17}D_4
% in TeX

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

G:=SmallGroup(320,814);
// by ID

G=gap.SmallGroup(320,814);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,758,135,184,570,297,136,12550]);
// Polycyclic

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

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