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G = D20.32D4order 320 = 26·5

2nd non-split extension by D20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.32D4, C23.11D20, Dic10.31D4, C22⋊C85D5, C4.121(D4×D5), D205C45C2, C10.9C22≀C2, (C2×Dic20)⋊2C2, C10.8(C4○D8), C20.333(C2×D4), (C2×C4).118D20, (C2×C20).240D4, (C2×C8).108D10, C51(D4.7D4), (C2×C40).3C22, (C22×C4).83D10, (C22×C10).53D4, C20.44D410C2, C20.48D416C2, (C2×C20).743C23, C22.106(C2×D20), C10.9(C8.C22), C4⋊Dic5.12C22, C2.12(C22⋊D20), C2.10(D407C2), C2.12(C8.D10), (C2×D20).198C22, (C22×C20).96C22, (C2×Dic10).216C22, (C5×C22⋊C8)⋊7C2, (C2×C40⋊C2)⋊10C2, (C2×C4○D20).2C2, (C2×C10).126(C2×D4), (C2×C4).688(C22×D5), SmallGroup(320,360)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.32D4
C1C5C10C20C2×C20C2×D20C2×C4○D20 — D20.32D4
C5C10C2×C20 — D20.32D4
C1C22C22×C4C22⋊C8

Generators and relations for D20.32D4
 G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a5c3 >

Subgroups: 686 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×7], Q8 [×5], C23, C23, D5 [×2], C10 [×3], C10, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], Dic5 [×4], C20 [×2], C20, D10 [×4], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C40 [×2], Dic10 [×2], Dic10 [×3], C4×D5 [×4], D20 [×2], D20, C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, D4.7D4, C40⋊C2 [×2], Dic20 [×2], C10.D4, C4⋊Dic5, C23.D5, C2×C40 [×2], C2×Dic10 [×2], C2×C4×D5, C2×D20, C4○D20 [×4], C2×C5⋊D4, C22×C20, C20.44D4, D205C4, C5×C22⋊C8, C2×C40⋊C2, C2×Dic20, C20.48D4, C2×C4○D20, D20.32D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8.C22, D20 [×2], C22×D5, D4.7D4, C2×D20, D4×D5 [×2], C22⋊D20, D407C2, C8.D10, D20.32D4

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444455888810···101010101020···202020202040···40
size1111420202222202040402244442···244442···244444···4

59 irreducible representations

dim1111111122222222222444
type+++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10C4○D8D20D20D407C2C8.C22D4×D5C8.D10
kernelD20.32D4C20.44D4D205C4C5×C22⋊C8C2×C40⋊C2C2×Dic20C20.48D4C2×C4○D20Dic10D20C2×C20C22×C10C22⋊C8C2×C8C22×C4C10C2×C4C23C2C10C4C2
# reps11111111221124244416144

Matrix representation of D20.32D4 in GL6(𝔽41)

1580000
23260000
0004000
001700
0000400
0000040
,
1580000
13260000
000100
001000
0000400
0000121
,
4140000
30130000
00303200
0091100
00002939
00001012
,
900000
090000
001000
000100
000010
00002940

G:=sub<GL(6,GF(41))| [15,23,0,0,0,0,8,26,0,0,0,0,0,0,0,1,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[15,13,0,0,0,0,8,26,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,12,0,0,0,0,0,1],[4,30,0,0,0,0,14,13,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,29,10,0,0,0,0,39,12],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,29,0,0,0,0,0,40] >;

D20.32D4 in GAP, Magma, Sage, TeX

D_{20}._{32}D_4
% in TeX

G:=Group("D20.32D4");
// GroupNames label

G:=SmallGroup(320,360);
// by ID

G=gap.SmallGroup(320,360);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,1123,136,12550]);
// Polycyclic

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

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