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G = D20.23D4order 320 = 26·5

6th non-split extension by D20 of D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.23D4, C42.63D10, C4.50(D4×D5), (C4×D20)⋊21C2, C203C829C2, C4.4D41D5, C20.24(C2×D4), (C2×D4).47D10, (C2×C20).271D4, C55(D4.2D4), (C2×Q8).37D10, C20.68(C4○D4), D4⋊Dic519C2, Q8⋊Dic522C2, C4.2(D42D5), C10.105(C4○D8), C2.11(C202D4), (C4×C20).106C22, (C2×C20).375C23, (D4×C10).63C22, (Q8×C10).55C22, C2.18(D4⋊D10), C10.102(C4⋊D4), C10.119(C8⋊C22), (C2×D20).252C22, C4⋊Dic5.341C22, C2.24(D4.8D10), (C2×Q8⋊D5)⋊12C2, (C2×D4⋊D5).6C2, (C5×C4.4D4)⋊1C2, (C2×C10).506(C2×D4), (C2×C4).61(C5⋊D4), (C2×C4).475(C22×D5), C22.181(C2×C5⋊D4), (C2×C52C8).121C22, SmallGroup(320,684)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.23D4
C1C5C10C20C2×C20C2×D20C4×D20 — D20.23D4
C5C10C2×C20 — D20.23D4
C1C22C42C4.4D4

Generators and relations for D20.23D4
 G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=a10c-1 >

Subgroups: 518 in 124 conjugacy classes, 41 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 [×5], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C10, 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 [×4], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, D4.2D4, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×2], Q8⋊D5 [×2], C4×C20, C5×C22⋊C4 [×2], C2×C4×D5, C2×D20, D4×C10, Q8×C10, C203C8, D4⋊Dic5, Q8⋊Dic5, C4×D20, C2×D4⋊D5, C2×Q8⋊D5, C5×C4.4D4, D20.23D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4.2D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, D4⋊D10, D4.8D10, D20.23D4

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222224444444455888810···101010101020···2020202020
size111182020222248202022202020202···288884···48888

47 irreducible representations

dim1111111122222222244444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C5⋊D4C8⋊C22D4×D5D42D5D4⋊D10D4.8D10
kernelD20.23D4C203C8D4⋊Dic5Q8⋊Dic5C4×D20C2×D4⋊D5C2×Q8⋊D5C5×C4.4D4D20C2×C20C4.4D4C20C42C2×D4C2×Q8C10C2×C4C10C4C4C2C2
# reps1111111122222224812244

Matrix representation of D20.23D4 in GL6(𝔽41)

35400000
100000
0004000
001000
000010
000001
,
0400000
4000000
0040000
000100
0000400
0000040
,
100000
010000
0032000
0003200
0000938
0000032
,
1860000
35230000
00151500
00152600
00003013
00001611

G:=sub<GL(6,GF(41))| [35,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,38,32],[18,35,0,0,0,0,6,23,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,30,16,0,0,0,0,13,11] >;

D20.23D4 in GAP, Magma, Sage, TeX

D_{20}._{23}D_4
% in TeX

G:=Group("D20.23D4");
// GroupNames label

G:=SmallGroup(320,684);
// by ID

G=gap.SmallGroup(320,684);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,1123,297,136,12550]);
// Polycyclic

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

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