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G = D4.Dic10order 320 = 26·5

1st non-split extension by D4 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.1Dic10, C4⋊C4.4D10, C51(D4.Q8), (C5×D4).1Q8, C20.3(C2×Q8), C406C411C2, D4⋊C4.6D5, (C2×C8).114D10, C10.D84C2, C4.Dic101C2, (D4×Dic5).5C2, C4.3(C2×Dic10), (C2×D4).128D10, C10.37(C4○D8), C20.8Q811C2, D4⋊Dic5.3C2, C22.165(D4×D5), C10.9(C22⋊Q8), C20.147(C4○D4), C4.76(D42D5), C2.10(D8⋊D5), C10.27(C8⋊C22), (C2×C40).125C22, (C2×C20).203C23, (C2×Dic5).193D4, (D4×C10).24C22, C4⋊Dic5.62C22, C2.8(SD163D5), (C4×Dic5).13C22, C2.14(Dic5.14D4), (C5×C4⋊C4).8C22, (C5×D4⋊C4).6C2, (C2×C10).216(C2×D4), (C2×C52C8).9C22, (C2×C4).310(C22×D5), SmallGroup(320,390)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.Dic10
C1C5C10C2×C10C2×C20C4×Dic5D4×Dic5 — D4.Dic10
C5C10C2×C20 — D4.Dic10
C1C22C2×C4D4⋊C4

Generators and relations for D4.Dic10
 G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd-1=a2c-1 >

Subgroups: 374 in 102 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8, C2×C8, C22×C4, C2×D4, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×4], D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C40, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, D4.Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C22×Dic5, D4×C10, C10.D8, C20.8Q8, C406C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, D4×Dic5, D4.Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8⋊C22, Dic10 [×2], C22×D5, D4.Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D8⋊D5, SD163D5, D4.Dic10

Smallest permutation representation of D4.Dic10
On 160 points
Generators in S160
(1 141 32 124)(2 125 33 142)(3 143 34 126)(4 127 35 144)(5 145 36 128)(6 129 37 146)(7 147 38 130)(8 131 39 148)(9 149 40 132)(10 133 21 150)(11 151 22 134)(12 135 23 152)(13 153 24 136)(14 137 25 154)(15 155 26 138)(16 139 27 156)(17 157 28 140)(18 121 29 158)(19 159 30 122)(20 123 31 160)(41 91 77 120)(42 101 78 92)(43 93 79 102)(44 103 80 94)(45 95 61 104)(46 105 62 96)(47 97 63 106)(48 107 64 98)(49 99 65 108)(50 109 66 100)(51 81 67 110)(52 111 68 82)(53 83 69 112)(54 113 70 84)(55 85 71 114)(56 115 72 86)(57 87 73 116)(58 117 74 88)(59 89 75 118)(60 119 76 90)
(1 134)(2 12)(3 136)(4 14)(5 138)(6 16)(7 140)(8 18)(9 122)(10 20)(11 124)(13 126)(15 128)(17 130)(19 132)(21 31)(22 141)(23 33)(24 143)(25 35)(26 145)(27 37)(28 147)(29 39)(30 149)(32 151)(34 153)(36 155)(38 157)(40 159)(41 67)(42 111)(43 69)(44 113)(45 71)(46 115)(47 73)(48 117)(49 75)(50 119)(51 77)(52 101)(53 79)(54 103)(55 61)(56 105)(57 63)(58 107)(59 65)(60 109)(62 86)(64 88)(66 90)(68 92)(70 94)(72 96)(74 98)(76 100)(78 82)(80 84)(81 91)(83 93)(85 95)(87 97)(89 99)(102 112)(104 114)(106 116)(108 118)(110 120)(121 148)(123 150)(125 152)(127 154)(129 156)(131 158)(133 160)(135 142)(137 144)(139 146)
(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)
(1 109 11 119)(2 99 12 89)(3 107 13 117)(4 97 14 87)(5 105 15 115)(6 95 16 85)(7 103 17 113)(8 93 18 83)(9 101 19 111)(10 91 20 81)(21 120 31 110)(22 90 32 100)(23 118 33 108)(24 88 34 98)(25 116 35 106)(26 86 36 96)(27 114 37 104)(28 84 38 94)(29 112 39 102)(30 82 40 92)(41 160 51 150)(42 122 52 132)(43 158 53 148)(44 140 54 130)(45 156 55 146)(46 138 56 128)(47 154 57 144)(48 136 58 126)(49 152 59 142)(50 134 60 124)(61 139 71 129)(62 155 72 145)(63 137 73 127)(64 153 74 143)(65 135 75 125)(66 151 76 141)(67 133 77 123)(68 149 78 159)(69 131 79 121)(70 147 80 157)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444455888810···1010101010202020202020202040···40
size111144228101020202040224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++-++++-+-+
imageC1C2C2C2C2C2C2C2D4Q8D5C4○D4D10D10D10C4○D8Dic10C8⋊C22D42D5D4×D5D8⋊D5SD163D5
kernelD4.Dic10C10.D8C20.8Q8C406C4D4⋊Dic5C5×D4⋊C4C4.Dic10D4×Dic5C2×Dic5C5×D4D4⋊C4C20C4⋊C4C2×C8C2×D4C10D4C10C4C22C2C2
# reps1111111122222224812244

Matrix representation of D4.Dic10 in GL4(𝔽41) generated by

11800
94000
0010
0001
,
402300
0100
00400
00040
,
111700
293000
003039
001614
,
32000
03200
001837
003023
G:=sub<GL(4,GF(41))| [1,9,0,0,18,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,23,1,0,0,0,0,40,0,0,0,0,40],[11,29,0,0,17,30,0,0,0,0,30,16,0,0,39,14],[32,0,0,0,0,32,0,0,0,0,18,30,0,0,37,23] >;

D4.Dic10 in GAP, Magma, Sage, TeX

D_4.{\rm Dic}_{10}
% in TeX

G:=Group("D4.Dic10");
// GroupNames label

G:=SmallGroup(320,390);
// by ID

G=gap.SmallGroup(320,390);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,926,219,58,851,438,102,12550]);
// Polycyclic

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

׿
×
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