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G = Dic56SD16order 320 = 26·5

1st semidirect product of Dic5 and SD16 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic56SD16, C53(C4×SD16), D4.D56C4, D4.2(C4×D5), C10.62(C4×D4), C2.1(D5×SD16), C4⋊C4.130D10, (C8×Dic5)⋊18C2, D4⋊C4.9D5, (C2×C8).198D10, C20.Q81C2, (D4×Dic5).2C2, C22.67(D4×D5), Dic1011(C2×C4), Dic53Q82C2, (C2×D4).125D10, C10.20(C4○D8), C2.1(D83D5), C20.38(C22×C4), C10.18(C2×SD16), C20.144(C4○D4), C20.44D415C2, C4.45(D42D5), (C2×C40).177C22, (C2×C20).198C23, (C2×Dic5).268D4, (D4×C10).19C22, C4⋊Dic5.58C22, C2.16(Dic54D4), (C2×Dic10).54C22, (C4×Dic5).252C22, C4.3(C2×C4×D5), C52C818(C2×C4), (C5×D4).16(C2×C4), (C5×C4⋊C4).3C22, (C2×D4.D5).2C2, (C5×D4⋊C4).8C2, (C2×C10).211(C2×D4), (C2×C4).305(C22×D5), (C2×C52C8).216C22, SmallGroup(320,385)

Series: Derived Chief Lower central Upper central

C1C20 — Dic56SD16
C1C5C10C20C2×C20C4×Dic5D4×Dic5 — Dic56SD16
C5C10C20 — Dic56SD16
C1C22C2×C4D4⋊C4

Generators and relations for Dic56SD16
 G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 422 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C5, C8 [×3], C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×3], C23, C10 [×3], C10 [×2], C42 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×4], C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C52C8 [×2], C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C4×SD16, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D4.D5 [×4], C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C22×Dic5, D4×C10, C20.Q8, C8×Dic5, C20.44D4, C5×D4⋊C4, Dic53Q8, C2×D4.D5, D4×Dic5, Dic56SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, SD16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×SD16, C4○D8, C4×D5 [×2], C22×D5, C4×SD16, C2×C4×D5, D4×D5, D42D5, Dic54D4, D83D5, D5×SD16, Dic56SD16

Smallest permutation representation of Dic56SD16
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)
(1 102 6 107)(2 101 7 106)(3 110 8 105)(4 109 9 104)(5 108 10 103)(11 75 16 80)(12 74 17 79)(13 73 18 78)(14 72 19 77)(15 71 20 76)(21 92 26 97)(22 91 27 96)(23 100 28 95)(24 99 29 94)(25 98 30 93)(31 121 36 126)(32 130 37 125)(33 129 38 124)(34 128 39 123)(35 127 40 122)(41 112 46 117)(42 111 47 116)(43 120 48 115)(44 119 49 114)(45 118 50 113)(51 141 56 146)(52 150 57 145)(53 149 58 144)(54 148 59 143)(55 147 60 142)(61 132 66 137)(62 131 67 136)(63 140 68 135)(64 139 69 134)(65 138 70 133)(81 159 86 154)(82 158 87 153)(83 157 88 152)(84 156 89 151)(85 155 90 160)
(1 160 40 140 28 16 48 147)(2 159 31 139 29 15 49 146)(3 158 32 138 30 14 50 145)(4 157 33 137 21 13 41 144)(5 156 34 136 22 12 42 143)(6 155 35 135 23 11 43 142)(7 154 36 134 24 20 44 141)(8 153 37 133 25 19 45 150)(9 152 38 132 26 18 46 149)(10 151 39 131 27 17 47 148)(51 101 86 121 69 94 71 114)(52 110 87 130 70 93 72 113)(53 109 88 129 61 92 73 112)(54 108 89 128 62 91 74 111)(55 107 90 127 63 100 75 120)(56 106 81 126 64 99 76 119)(57 105 82 125 65 98 77 118)(58 104 83 124 66 97 78 117)(59 103 84 123 67 96 79 116)(60 102 85 122 68 95 80 115)
(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 141)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(111 128)(112 129)(113 130)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(120 127)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444444444558888888810···1010101010202020202020202040···40
size11114422445555101020202020222222101010102···28888444488884···4

56 irreducible representations

dim1111111112222222224444
type+++++++++++++-+-
imageC1C2C2C2C2C2C2C2C4D4D5SD16C4○D4D10D10D10C4○D8C4×D5D42D5D4×D5D83D5D5×SD16
kernelDic56SD16C20.Q8C8×Dic5C20.44D4C5×D4⋊C4Dic53Q8C2×D4.D5D4×Dic5D4.D5C2×Dic5D4⋊C4Dic5C20C4⋊C4C2×C8C2×D4C10D4C4C22C2C2
# reps1111111182242222482244

Matrix representation of Dic56SD16 in GL6(𝔽41)

4000000
0400000
001000
000100
0000401
0000535
,
900000
090000
001000
000100
00001226
00001529
,
15150000
26150000
0001300
00193000
00001226
00001529
,
100000
0400000
0012100
0004000
000010
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,5,0,0,0,0,1,35],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,15,0,0,0,0,26,29],[15,26,0,0,0,0,15,15,0,0,0,0,0,0,0,19,0,0,0,0,13,30,0,0,0,0,0,0,12,15,0,0,0,0,26,29],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,21,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic56SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("Dic5:6SD16");
// GroupNames label

G:=SmallGroup(320,385);
// by ID

G=gap.SmallGroup(320,385);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,135,268,570,297,136,12550]);
// Polycyclic

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

׿
×
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