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## G = Dic5⋊3SD16order 320 = 26·5

### 2nd semidirect product of Dic5 and SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5⋊3SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — D4×Dic5 — Dic5⋊3SD16
 Lower central C5 — C10 — C2×C20 — Dic5⋊3SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for Dic53SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c3 >

Subgroups: 438 in 120 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×4], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20, C2×C10, C2×C10 [×4], Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C2×SD16, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C22×C10, D4.D4, C2×C52C8, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D4.D5 [×2], C23.D5, C2×C40, C5×SD16 [×2], C2×Dic10, C22×Dic5, D4×C10, Q8×C10, C20.8Q8, C20.44D4, Q8⋊Dic5, C2×D4.D5, D4×Dic5, Dic5⋊Q8, C10×SD16, Dic53SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8.C22, C5⋊D4 [×2], C22×D5, D4.D4, D4×D5, D42D5, C2×C5⋊D4, D5×SD16, SD16⋊D5, Dic5⋊D4, Dic53SD16

Smallest permutation representation of Dic53SD16
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 95 6 100)(2 94 7 99)(3 93 8 98)(4 92 9 97)(5 91 10 96)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(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)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
(1 85 40 68 28 80 48 60)(2 86 31 69 29 71 49 51)(3 87 32 70 30 72 50 52)(4 88 33 61 21 73 41 53)(5 89 34 62 22 74 42 54)(6 90 35 63 23 75 43 55)(7 81 36 64 24 76 44 56)(8 82 37 65 25 77 45 57)(9 83 38 66 26 78 46 58)(10 84 39 67 27 79 47 59)(11 120 142 107 155 127 135 100)(12 111 143 108 156 128 136 91)(13 112 144 109 157 129 137 92)(14 113 145 110 158 130 138 93)(15 114 146 101 159 121 139 94)(16 115 147 102 160 122 140 95)(17 116 148 103 151 123 131 96)(18 117 149 104 152 124 132 97)(19 118 150 105 153 125 133 98)(20 119 141 106 154 126 134 99)
(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), (1,95,6,100)(2,94,7,99)(3,93,8,98)(4,92,9,97)(5,91,10,96)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(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)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155), (1,85,40,68,28,80,48,60)(2,86,31,69,29,71,49,51)(3,87,32,70,30,72,50,52)(4,88,33,61,21,73,41,53)(5,89,34,62,22,74,42,54)(6,90,35,63,23,75,43,55)(7,81,36,64,24,76,44,56)(8,82,37,65,25,77,45,57)(9,83,38,66,26,78,46,58)(10,84,39,67,27,79,47,59)(11,120,142,107,155,127,135,100)(12,111,143,108,156,128,136,91)(13,112,144,109,157,129,137,92)(14,113,145,110,158,130,138,93)(15,114,146,101,159,121,139,94)(16,115,147,102,160,122,140,95)(17,116,148,103,151,123,131,96)(18,117,149,104,152,124,132,97)(19,118,150,105,153,125,133,98)(20,119,141,106,154,126,134,99), (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), (1,95,6,100)(2,94,7,99)(3,93,8,98)(4,92,9,97)(5,91,10,96)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(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)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155), (1,85,40,68,28,80,48,60)(2,86,31,69,29,71,49,51)(3,87,32,70,30,72,50,52)(4,88,33,61,21,73,41,53)(5,89,34,62,22,74,42,54)(6,90,35,63,23,75,43,55)(7,81,36,64,24,76,44,56)(8,82,37,65,25,77,45,57)(9,83,38,66,26,78,46,58)(10,84,39,67,27,79,47,59)(11,120,142,107,155,127,135,100)(12,111,143,108,156,128,136,91)(13,112,144,109,157,129,137,92)(14,113,145,110,158,130,138,93)(15,114,146,101,159,121,139,94)(16,115,147,102,160,122,140,95)(17,116,148,103,151,123,131,96)(18,117,149,104,152,124,132,97)(19,118,150,105,153,125,133,98)(20,119,141,106,154,126,134,99), (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)], [(1,95,6,100),(2,94,7,99),(3,93,8,98),(4,92,9,97),(5,91,10,96),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(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),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)], [(1,85,40,68,28,80,48,60),(2,86,31,69,29,71,49,51),(3,87,32,70,30,72,50,52),(4,88,33,61,21,73,41,53),(5,89,34,62,22,74,42,54),(6,90,35,63,23,75,43,55),(7,81,36,64,24,76,44,56),(8,82,37,65,25,77,45,57),(9,83,38,66,26,78,46,58),(10,84,39,67,27,79,47,59),(11,120,142,107,155,127,135,100),(12,111,143,108,156,128,136,91),(13,112,144,109,157,129,137,92),(14,113,145,110,158,130,138,93),(15,114,146,101,159,121,139,94),(16,115,147,102,160,122,140,95),(17,116,148,103,151,123,131,96),(18,117,149,104,152,124,132,97),(19,118,150,105,153,125,133,98),(20,119,141,106,154,126,134,99)], [(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)])

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 2 2 8 10 10 20 20 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 SD16 C4○D4 D10 D10 D10 C5⋊D4 C8.C22 D4⋊2D5 D4×D5 D5×SD16 SD16⋊D5 kernel Dic5⋊3SD16 C20.8Q8 C20.44D4 Q8⋊Dic5 C2×D4.D5 D4×Dic5 Dic5⋊Q8 C10×SD16 C2×Dic5 C5×D4 C2×SD16 Dic5 C20 C2×C8 C2×D4 C2×Q8 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 2 4 4

Matrix representation of Dic53SD16 in GL6(𝔽41)

 40 1 0 0 0 0 5 35 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 18 21 0 0 0 0 10 23 0 0 0 0 0 0 32 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 10 2 0 0 0 0 26 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 16 40

G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,10,0,0,0,0,21,23,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,10,26,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,16,0,0,0,0,0,40] >;

Dic53SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_3{\rm SD}_{16}
% in TeX

G:=Group("Dic5:3SD16");
// GroupNames label

G:=SmallGroup(320,789);
// by ID

G=gap.SmallGroup(320,789);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,135,184,570,297,136,12550]);
// Polycyclic

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

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