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## G = D10⋊SD16order 320 = 26·5

### 1st semidirect product of D10 and SD16 acting via SD16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10⋊SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C2×C8 — D10⋊SD16
 Lower central C5 — C10 — C2×C20 — D10⋊SD16
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D10⋊SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a8b, dbd=a5b, dcd=c3 >

Subgroups: 518 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×3], C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C10, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8 [×3], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8 [×2], C40, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C88D4, C8×D5 [×2], C2×C52C8, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D4.D5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×C5⋊D4, D4×C10, C20.Q8, C20.44D4, C5×D4⋊C4, D102Q8, D5×C2×C8, C2×D4.D5, C202D4, D10⋊SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, C22×D5, C88D4, C4○D20, D4×D5 [×2], D10⋊D4, D83D5, D5×SD16, D10⋊SD16

Smallest permutation representation of D10⋊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 10)(2 9)(3 8)(4 7)(5 6)(11 18)(12 17)(13 16)(14 15)(19 20)(21 24)(22 23)(25 30)(26 29)(27 28)(31 35)(32 34)(36 40)(37 39)(41 43)(44 50)(45 49)(46 48)(51 58)(52 57)(53 56)(54 55)(59 60)(62 70)(63 69)(64 68)(65 67)(71 79)(72 78)(73 77)(74 76)(81 88)(82 87)(83 86)(84 85)(89 90)(91 93)(94 100)(95 99)(96 98)(101 106)(102 105)(103 104)(107 110)(108 109)(111 114)(112 113)(115 120)(116 119)(117 118)(121 123)(124 130)(125 129)(126 128)(131 138)(132 137)(133 136)(134 135)(139 140)(142 150)(143 149)(144 148)(145 147)(151 159)(152 158)(153 157)(154 156)
(1 118 28 135 55 90 15 104)(2 117 29 134 56 89 16 103)(3 116 30 133 57 88 17 102)(4 115 21 132 58 87 18 101)(5 114 22 131 59 86 19 110)(6 113 23 140 60 85 20 109)(7 112 24 139 51 84 11 108)(8 111 25 138 52 83 12 107)(9 120 26 137 53 82 13 106)(10 119 27 136 54 81 14 105)(31 95 153 78 149 64 125 50)(32 94 154 77 150 63 126 49)(33 93 155 76 141 62 127 48)(34 92 156 75 142 61 128 47)(35 91 157 74 143 70 129 46)(36 100 158 73 144 69 130 45)(37 99 159 72 145 68 121 44)(38 98 160 71 146 67 122 43)(39 97 151 80 147 66 123 42)(40 96 152 79 148 65 124 41)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 61)(9 62)(10 63)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 118)(32 119)(33 120)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 97)(58 98)(59 99)(60 100)(81 150)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(101 160)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 SD16 D10 D10 D10 C4○D8 C4○D20 D4×D5 D4×D5 D8⋊3D5 D5×SD16 kernel D10⋊SD16 C20.Q8 C20.44D4 C5×D4⋊C4 D10⋊2Q8 D5×C2×C8 C2×D4.D5 C20⋊2D4 C5⋊2C8 C2×Dic5 C22×D5 D4⋊C4 C20 D10 C4⋊C4 C2×C8 C2×D4 C10 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 4 2 2 2 4 8 2 2 4 4

Matrix representation of D10⋊SD16 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 34 34 0 0 7 1
,
 1 0 0 0 0 1 0 0 0 0 34 34 0 0 1 7
,
 38 0 0 0 11 14 0 0 0 0 9 0 0 0 19 32
,
 37 5 0 0 38 4 0 0 0 0 17 40 0 0 1 24
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,34,7,0,0,34,1],[1,0,0,0,0,1,0,0,0,0,34,1,0,0,34,7],[38,11,0,0,0,14,0,0,0,0,9,19,0,0,0,32],[37,38,0,0,5,4,0,0,0,0,17,1,0,0,40,24] >;

D10⋊SD16 in GAP, Magma, Sage, TeX

D_{10}\rtimes {\rm SD}_{16}
% in TeX

G:=Group("D10:SD16");
// GroupNames label

G:=SmallGroup(320,405);
// by ID

G=gap.SmallGroup(320,405);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,297,136,438,102,12550]);
// Polycyclic

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

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