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G = D45Dic10order 320 = 26·5

1st semidirect product of D4 and Dic10 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D45Dic10, C42.103D10, C10.132+ 1+4, (C5×D4)⋊6Q8, C20⋊Q815C2, C52(D43Q8), (C4×D4).11D5, C20.42(C2×Q8), C4⋊C4.278D10, (D4×C20).12C2, (C4×Dic10)⋊26C2, (C2×D4).242D10, C20.48D47C2, (C2×C10).83C24, C20.6Q814C2, C4.Dic1014C2, (D4×Dic5).12C2, C4.15(C2×Dic10), C10.13(C22×Q8), (C4×C20).146C22, (C2×C20).154C23, C22⋊C4.106D10, (C22×C4).202D10, C4⋊Dic5.37C22, C2.16(D46D10), C22.1(C2×Dic10), Dic5.35(C4○D4), Dic5.14D47C2, C23.D5.8C22, (D4×C10).249C22, (C22×C20).77C22, (C2×Dic5).33C23, (C4×Dic5).80C22, C2.15(C22×Dic10), C23.163(C22×D5), C22.111(C23×D5), (C22×C10).153C23, (C2×Dic10).27C22, C10.D4.108C22, (C22×Dic5).91C22, C2.18(D5×C4○D4), (C2×C10).3(C2×Q8), C10.137(C2×C4○D4), (C2×C10.D4)⋊24C2, (C5×C4⋊C4).319C22, (C2×C4).154(C22×D5), (C5×C22⋊C4).104C22, SmallGroup(320,1211)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D45Dic10
C1C5C10C2×C10C2×Dic5C22×Dic5D4×Dic5 — D45Dic10
C5C2×C10 — D45Dic10
C1C22C4×D4

Generators and relations for D45Dic10
 G = < a,b,c,d | a4=b2=c20=1, d2=c10, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >

Subgroups: 694 in 228 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×4], C22 [×4], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×4], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×C10 [×2], D43Q8, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×8], C4⋊Dic5 [×3], C4⋊Dic5 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C22×Dic5 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C20.6Q8, Dic5.14D4 [×4], C20⋊Q8, C4.Dic10, C2×C10.D4 [×2], C20.48D4 [×2], D4×Dic5 [×2], D4×C20, D45Dic10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ 1+4, Dic10 [×4], C22×D5 [×7], D43Q8, C2×Dic10 [×6], C23×D5, C22×Dic10, D46D10, D5×C4○D4, D45Dic10

Smallest permutation representation of D45Dic10
On 160 points
Generators in S160
(1 24 140 94)(2 95 121 25)(3 26 122 96)(4 97 123 27)(5 28 124 98)(6 99 125 29)(7 30 126 100)(8 81 127 31)(9 32 128 82)(10 83 129 33)(11 34 130 84)(12 85 131 35)(13 36 132 86)(14 87 133 37)(15 38 134 88)(16 89 135 39)(17 40 136 90)(18 91 137 21)(19 22 138 92)(20 93 139 23)(41 62 108 159)(42 160 109 63)(43 64 110 141)(44 142 111 65)(45 66 112 143)(46 144 113 67)(47 68 114 145)(48 146 115 69)(49 70 116 147)(50 148 117 71)(51 72 118 149)(52 150 119 73)(53 74 120 151)(54 152 101 75)(55 76 102 153)(56 154 103 77)(57 78 104 155)(58 156 105 79)(59 80 106 157)(60 158 107 61)
(1 34)(2 85)(3 36)(4 87)(5 38)(6 89)(7 40)(8 91)(9 22)(10 93)(11 24)(12 95)(13 26)(14 97)(15 28)(16 99)(17 30)(18 81)(19 32)(20 83)(21 127)(23 129)(25 131)(27 133)(29 135)(31 137)(33 139)(35 121)(37 123)(39 125)(41 149)(42 73)(43 151)(44 75)(45 153)(46 77)(47 155)(48 79)(49 157)(50 61)(51 159)(52 63)(53 141)(54 65)(55 143)(56 67)(57 145)(58 69)(59 147)(60 71)(62 118)(64 120)(66 102)(68 104)(70 106)(72 108)(74 110)(76 112)(78 114)(80 116)(82 138)(84 140)(86 122)(88 124)(90 126)(92 128)(94 130)(96 132)(98 134)(100 136)(101 142)(103 144)(105 146)(107 148)(109 150)(111 152)(113 154)(115 156)(117 158)(119 160)
(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 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 151 31 141)(22 150 32 160)(23 149 33 159)(24 148 34 158)(25 147 35 157)(26 146 36 156)(27 145 37 155)(28 144 38 154)(29 143 39 153)(30 142 40 152)(61 94 71 84)(62 93 72 83)(63 92 73 82)(64 91 74 81)(65 90 75 100)(66 89 76 99)(67 88 77 98)(68 87 78 97)(69 86 79 96)(70 85 80 95)(101 126 111 136)(102 125 112 135)(103 124 113 134)(104 123 114 133)(105 122 115 132)(106 121 116 131)(107 140 117 130)(108 139 118 129)(109 138 119 128)(110 137 120 127)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L···4Q5A5B10A···10F10G···10N20A···20H20I···20X
order12222222444444444444···45510···1010···1020···2020···20
size1111222222224441010101020···20222···24···42···24···4

65 irreducible representations

dim1111111111222222222444
type++++++++++-++++++-+
imageC1C2C2C2C2C2C2C2C2C2Q8D5C4○D4D10D10D10D10D10Dic102+ 1+4D46D10D5×C4○D4
kernelD45Dic10C4×Dic10C20.6Q8Dic5.14D4C20⋊Q8C4.Dic10C2×C10.D4C20.48D4D4×Dic5D4×C20C5×D4C4×D4Dic5C42C22⋊C4C4⋊C4C22×C4C2×D4D4C10C2C2
# reps11141122214242424216144

Matrix representation of D45Dic10 in GL6(𝔽41)

4000000
0400000
001000
000100
0000121
00003740
,
100000
010000
001000
000100
0000121
0000040
,
34310000
570000
0014000
0083400
0000320
0000369
,
40110000
1110000
003500
00233800
0000121
00003740

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[34,5,0,0,0,0,31,7,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,32,36,0,0,0,0,0,9],[40,11,0,0,0,0,11,1,0,0,0,0,0,0,3,23,0,0,0,0,5,38,0,0,0,0,0,0,1,37,0,0,0,0,21,40] >;

D45Dic10 in GAP, Magma, Sage, TeX

D_4\rtimes_5{\rm Dic}_{10}
% in TeX

G:=Group("D4:5Dic10");
// GroupNames label

G:=SmallGroup(320,1211);
// by ID

G=gap.SmallGroup(320,1211);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,675,192,12550]);
// 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=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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