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G = SD16×Dic5order 320 = 26·5

Direct product of SD16 and Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD16×Dic5, C58(C4×SD16), C4025(C2×C4), C85(C2×Dic5), Q81(C2×Dic5), (Q8×Dic5)⋊3C2, (C5×SD16)⋊7C4, (C8×Dic5)⋊9C2, C406C426C2, C2.7(D5×SD16), (C2×C8).261D10, C10.125(C4×D4), (D4×Dic5).7C2, D4.1(C2×Dic5), (C2×SD16).5D5, C2.12(D4×Dic5), (C2×D4).142D10, C20.95(C4○D4), C10.59(C4○D8), Q8⋊Dic524C2, (C2×Q8).112D10, (C10×SD16).3C2, C10.42(C2×SD16), C22.116(D4×D5), C4.30(D42D5), C4.3(C22×Dic5), (C2×C20).438C23, C20.132(C22×C4), (C2×C40).162C22, (C2×Dic5).281D4, D4⋊Dic5.14C2, (D4×C10).87C22, (Q8×C10).68C22, C2.7(SD163D5), C4⋊Dic5.168C22, (C4×Dic5).271C22, (C5×Q8)⋊15(C2×C4), (C5×D4).22(C2×C4), (C2×C10).350(C2×D4), (C2×C4).527(C22×D5), (C2×C52C8).280C22, SmallGroup(320,788)

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, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 390 in 122 conjugacy classes, 59 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], 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 [×2], Dic5 [×3], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, 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, C4×SD16, C2×C52C8, C4×Dic5, C4×Dic5, C4⋊Dic5 [×2], C4⋊Dic5, C23.D5, C2×C40, C5×SD16 [×4], C22×Dic5, D4×C10, Q8×C10, C8×Dic5, C406C4, 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, SD16 [×2], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×SD16, C4○D8, C2×Dic5 [×6], C22×D5, C4×SD16, D4×D5, D42D5, C22×Dic5, D5×SD16, SD163D5, D4×Dic5, SD16×Dic5

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim1111111112222222224444
type+++++++++++-++-+
imageC1C2C2C2C2C2C2C2C4D4D5SD16C4○D4D10Dic5D10D10C4○D8D42D5D4×D5D5×SD16SD163D5
kernelSD16×Dic5C8×Dic5C406C4D4⋊Dic5Q8⋊Dic5D4×Dic5Q8×Dic5C10×SD16C5×SD16C2×Dic5C2×SD16Dic5C20C2×C8SD16C2×D4C2×Q8C10C4C22C2C2
# reps1111111182242282242244

Matrix representation of SD16×Dic5 in GL4(𝔽41) generated by

02600
303000
00400
00040
,
1100
04000
0010
0001
,
40000
04000
00140
00366
,
32000
03200
00932
00032
G:=sub<GL(4,GF(41))| [0,30,0,0,26,30,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,36,0,0,40,6],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,32,32] >;

SD16×Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("SD16xDic5");
// GroupNames label

G:=SmallGroup(320,788);
// by ID

G=gap.SmallGroup(320,788);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,851,438,102,12550]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽