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G = Dic5.SD16order 320 = 26·5

6th non-split extension by Dic5 of SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.4D8, Dic5.6SD16, C5⋊(C4.D8), (C2×D4).3F5, (D4×C10).3C4, (C2×D20).5C4, C20⋊D4.3C2, Dic5⋊C81C2, C2.5(C23.F5), C2.15(D20⋊C4), (C2×Dic5).108D4, C10.4(C4.D4), C10.15(D4⋊C4), (C4×Dic5).4C22, C22.61(C22⋊F5), (C2×C4).16(C2×F5), (C2×C20).13(C2×C4), (C2×C10).38(C22⋊C4), SmallGroup(320,263)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic5.SD16
C1C5C10C2×C10C2×Dic5C4×Dic5Dic5⋊C8 — Dic5.SD16
C5C2×C10C2×C20 — Dic5.SD16
C1C22C2×C4C2×D4

Generators and relations for Dic5.SD16
 G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, cac-1=a3, ad=da, cbc-1=dbd=a5b, dcd=bc3 >

Subgroups: 490 in 84 conjugacy classes, 26 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], D5, C10, C10 [×2], C10, C42, C2×C8 [×2], C2×D4, C2×D4 [×3], Dic5 [×4], C20, D10 [×3], C2×C10, C2×C10 [×3], C4⋊C8 [×2], C41D4, C5⋊C8 [×2], D20, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4, C22×D5, C22×C10, C4.D8, C4×Dic5, C2×C5⋊C8 [×2], C2×D20, C2×C5⋊D4 [×2], D4×C10, Dic5⋊C8 [×2], C20⋊D4, Dic5.SD16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8 [×2], SD16 [×2], F5, C4.D4, D4⋊C4 [×2], C2×F5, C4.D8, C22⋊F5, D20⋊C4 [×2], C23.F5, Dic5.SD16

Character table of Dic5.SD16

 class 12A2B2C2D2E4A4B4C4D4E58A8B8C8D8E8F8G8H10A10B10C10D10E10F10G20A20B
 size 111184041010101042020202020202020444888888
ρ111111111111111111111111111111    trivial
ρ21111-1-11111111-1-1-1-1111111-1-1-1-111    linear of order 2
ρ31111-1-1111111-11111-1-1-1111-1-1-1-111    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1111111111    linear of order 2
ρ511111-11-1-1-1-11-iii-i-iii-i111111111    linear of order 4
ρ61111-111-1-1-1-11-i-i-iiiii-i111-1-1-1-111    linear of order 4
ρ71111-111-1-1-1-11iii-i-i-i-ii111-1-1-1-111    linear of order 4
ρ811111-11-1-1-1-11i-i-iii-i-ii111111111    linear of order 4
ρ9222200-22-2-222000000002220000-2-2    orthogonal lifted from D4
ρ10222200-2-222-22000000002220000-2-2    orthogonal lifted from D4
ρ112-22-200002-202-200002-222-2-2000000    orthogonal lifted from D8
ρ122-2-22000200-2202-2-22000-22-2000000    orthogonal lifted from D8
ρ132-2-22000200-220-222-2000-22-2000000    orthogonal lifted from D8
ρ142-22-200002-20220000-22-22-2-2000000    orthogonal lifted from D8
ρ152-2-22000-200220-2--2-2--2000-22-2000000    complex lifted from SD16
ρ162-2-22000-200220--2-2--2-2000-22-2000000    complex lifted from SD16
ρ172-22-20000-2202-20000-2--2--22-2-2000000    complex lifted from SD16
ρ182-22-20000-2202--20000--2-2-22-2-2000000    complex lifted from SD16
ρ1944444040000-100000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ2044-4-40000000400000000-4-44000000    orthogonal lifted from C4.D4
ρ214444-4040000-100000000-1-1-11111-1-1    orthogonal lifted from C2×F5
ρ22444400-40000-100000000-1-1-1-5-55511    orthogonal lifted from C22⋊F5
ρ23444400-40000-100000000-1-1-155-5-511    orthogonal lifted from C22⋊F5
ρ2444-4-40000000-10000000011-153+2ζ5+154+2ζ52+152+2ζ5+154+2ζ53+15-5    complex lifted from C23.F5
ρ2544-4-40000000-10000000011-152+2ζ5+154+2ζ53+154+2ζ52+153+2ζ5+1-55    complex lifted from C23.F5
ρ2644-4-40000000-10000000011-154+2ζ52+153+2ζ5+154+2ζ53+152+2ζ5+15-5    complex lifted from C23.F5
ρ2744-4-40000000-10000000011-154+2ζ53+152+2ζ5+153+2ζ5+154+2ζ52+1-55    complex lifted from C23.F5
ρ288-88-80000000-200000000-222000000    orthogonal lifted from D20⋊C4, Schur index 2
ρ298-8-880000000-2000000002-22000000    orthogonal lifted from D20⋊C4, Schur index 2

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

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

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

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

Matrix representation of Dic5.SD16 in GL6(𝔽41)

4000000
0400000
0004000
0000400
0000040
001111
,
0400000
100000
0027132232
002736514
009192814
001019532
,
26150000
15150000
0021201316
003437211
00255438
0040333620
,
4000000
010000
003631922
001914928
0013322722
001932105

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,1,0,0,0,40,0,1,0,0,0,0,40,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,27,27,9,10,0,0,13,36,19,19,0,0,22,5,28,5,0,0,32,14,14,32],[26,15,0,0,0,0,15,15,0,0,0,0,0,0,21,34,25,40,0,0,20,37,5,33,0,0,13,21,4,36,0,0,16,1,38,20],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,36,19,13,19,0,0,31,14,32,32,0,0,9,9,27,10,0,0,22,28,22,5] >;

Dic5.SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,263);
// by ID

G=gap.SmallGroup(320,263);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,100,1571,570,136,6278,3156]);
// Polycyclic

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

Export

Character table of Dic5.SD16 in TeX

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