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G = SD16⋊Dic5order 320 = 26·5

1st semidirect product of SD16 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD161Dic5, C4019(C2×C4), C408C44C2, C83(C2×Dic5), Q82(C2×Dic5), (C5×SD16)⋊5C4, (Q8×Dic5)⋊5C2, C405C426C2, (C2×C8).88D10, C10.126(C4×D4), (C2×SD16).1D5, (D4×Dic5).9C2, D4.2(C2×Dic5), C2.13(D4×Dic5), (C2×D4).144D10, C20.98(C4○D4), Q8⋊Dic526C2, C58(SD16⋊C4), C2.7(D40⋊C2), (C2×Q8).114D10, (C10×SD16).1C2, C22.117(D4×D5), C4.31(D42D5), C4.4(C22×Dic5), C10.76(C8⋊C22), C20.133(C22×C4), (C2×C40).113C22, (C2×C20).441C23, (C2×Dic5).239D4, D4⋊Dic5.15C2, C2.7(SD16⋊D5), (D4×C10).90C22, (Q8×C10).71C22, C10.46(C8.C22), C4⋊Dic5.171C22, (C4×Dic5).54C22, (C5×Q8)⋊16(C2×C4), (C5×D4).23(C2×C4), (C2×C10).353(C2×D4), (C2×C4).530(C22×D5), (C2×C52C8).153C22, SmallGroup(320,791)

Series: Derived Chief Lower central Upper central

C1C20 — SD16⋊Dic5
C1C5C10C2×C10C2×C20C4×Dic5D4×Dic5 — SD16⋊Dic5
C5C10C20 — SD16⋊Dic5
C1C22C2×C4C2×SD16

Generators and relations for SD16⋊Dic5
 G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 390 in 120 conjugacy classes, 57 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×2], Q8, C23, C10 [×3], C10 [×2], C42 [×2], C22⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C52C8, C40 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×C10, SD16⋊C4, C2×C52C8, C4×Dic5, C4×Dic5, C4⋊Dic5 [×2], C4⋊Dic5, C23.D5, C2×C40, C5×SD16 [×4], C22×Dic5, D4×C10, Q8×C10, C408C4, C405C4, D4⋊Dic5, Q8⋊Dic5, D4×Dic5, Q8×Dic5, C10×SD16, SD16⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C8⋊C22, C8.C22, C2×Dic5 [×6], C22×D5, SD16⋊C4, D4×D5, D42D5, C22×Dic5, D40⋊C2, SD16⋊D5, D4×Dic5, SD16⋊Dic5

Smallest permutation representation of SD16⋊Dic5
On 160 points
Generators in S160
(1 85 20 79 39 64 48 58)(2 86 11 80 40 65 49 59)(3 87 12 71 31 66 50 60)(4 88 13 72 32 67 41 51)(5 89 14 73 33 68 42 52)(6 90 15 74 34 69 43 53)(7 81 16 75 35 70 44 54)(8 82 17 76 36 61 45 55)(9 83 18 77 37 62 46 56)(10 84 19 78 38 63 47 57)(21 122 156 92 141 101 132 116)(22 123 157 93 142 102 133 117)(23 124 158 94 143 103 134 118)(24 125 159 95 144 104 135 119)(25 126 160 96 145 105 136 120)(26 127 151 97 146 106 137 111)(27 128 152 98 147 107 138 112)(28 129 153 99 148 108 139 113)(29 130 154 100 149 109 140 114)(30 121 155 91 150 110 131 115)
(11 49)(12 50)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 156)(22 157)(23 158)(24 159)(25 160)(26 151)(27 152)(28 153)(29 154)(30 155)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 121)(131 150)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
(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 92 6 97)(2 91 7 96)(3 100 8 95)(4 99 9 94)(5 98 10 93)(11 110 16 105)(12 109 17 104)(13 108 18 103)(14 107 19 102)(15 106 20 101)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444444455888810···1010101010202020202020202040···40
size11114422441010101020202020224420202···28888444488884···4

50 irreducible representations

dim1111111112222222444444
type+++++++++++-+++--++-
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10Dic5D10D10C8⋊C22C8.C22D42D5D4×D5D40⋊C2SD16⋊D5
kernelSD16⋊Dic5C408C4C405C4D4⋊Dic5Q8⋊Dic5D4×Dic5Q8×Dic5C10×SD16C5×SD16C2×Dic5C2×SD16C20C2×C8SD16C2×D4C2×Q8C10C10C4C22C2C2
# reps1111111182222822112244

Matrix representation of SD16⋊Dic5 in GL6(𝔽41)

100000
010000
0011166
00334011
0072600
0012666
,
4000000
0400000
001000
000100
003823400
003823040
,
35400000
100000
0014000
0036600
0010040
00354017
,
14390000
37270000
00131900
0042800
002201919
003219922

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,7,1,0,0,11,34,26,26,0,0,6,0,0,6,0,0,6,11,0,6],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,38,38,0,0,0,1,23,23,0,0,0,0,40,0,0,0,0,0,0,40],[35,1,0,0,0,0,40,0,0,0,0,0,0,0,1,36,1,35,0,0,40,6,0,40,0,0,0,0,0,1,0,0,0,0,40,7],[14,37,0,0,0,0,39,27,0,0,0,0,0,0,13,4,22,32,0,0,19,28,0,19,0,0,0,0,19,9,0,0,0,0,19,22] >;

SD16⋊Dic5 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes {\rm Dic}_5
% in TeX

G:=Group("SD16:Dic5");
// GroupNames label

G:=SmallGroup(320,791);
// by ID

G=gap.SmallGroup(320,791);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,184,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=c^5,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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