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## G = D10.16SD16order 320 = 26·5

### 1st non-split extension by D10 of SD16 acting via SD16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10.16SD16
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — D5×C4⋊C4 — D10.16SD16
 Lower central C5 — C10 — C2×C20 — D10.16SD16
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D10.16SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=a5c3 >

Subgroups: 494 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C22×C4 [×2], C2×D4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8 [×2], C2×C4⋊C4, C4⋊D4, C52C8, C40, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C23.46D4, C2×C52C8, C10.D4, C4⋊Dic5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×C5⋊D4, D4×C10, C20.Q8, C406C4, D101C8, D4⋊Dic5, C5×D4⋊C4, D5×C4⋊C4, C202D4, D10.16SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8⋊C22, C22×D5, C23.46D4, C4○D20, D4×D5, D42D5, D10.12D4, D8⋊D5, D5×SD16, D10.16SD16

Smallest permutation representation of D10.16SD16
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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)(21 134)(22 133)(23 132)(24 131)(25 140)(26 139)(27 138)(28 137)(29 136)(30 135)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 50)(38 49)(39 48)(40 47)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 90)(60 89)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(91 123)(92 122)(93 121)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(110 120)(141 154)(142 153)(143 152)(144 151)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)
(1 135 38 148 17 26 50 153)(2 136 39 149 18 27 41 154)(3 137 40 150 19 28 42 155)(4 138 31 141 20 29 43 156)(5 139 32 142 11 30 44 157)(6 140 33 143 12 21 45 158)(7 131 34 144 13 22 46 159)(8 132 35 145 14 23 47 160)(9 133 36 146 15 24 48 151)(10 134 37 147 16 25 49 152)(51 119 79 126 86 104 64 91)(52 120 80 127 87 105 65 92)(53 111 71 128 88 106 66 93)(54 112 72 129 89 107 67 94)(55 113 73 130 90 108 68 95)(56 114 74 121 81 109 69 96)(57 115 75 122 82 110 70 97)(58 116 76 123 83 101 61 98)(59 117 77 124 84 102 62 99)(60 118 78 125 85 103 63 100)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 71)(10 72)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 61)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 111)(30 112)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 151)(99 152)(100 153)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 131)(110 132)(121 144)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 141)(129 142)(130 143)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 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 2 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 8 10 10 2 2 4 4 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 C4○D4 SD16 D10 D10 D10 C4○D20 C8⋊C22 D4⋊2D5 D4×D5 D8⋊D5 D5×SD16 kernel D10.16SD16 C20.Q8 C40⋊6C4 D10⋊1C8 D4⋊Dic5 C5×D4⋊C4 D5×C4⋊C4 C20⋊2D4 C2×Dic5 C22×D5 D4⋊C4 C20 D10 C4⋊C4 C2×C8 C2×D4 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 4 4 2 2 2 8 1 2 2 4 4

Matrix representation of D10.16SD16 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 34 34 0 0 7 1
,
 40 0 0 0 0 40 0 0 0 0 1 7 0 0 0 40
,
 15 26 0 0 15 15 0 0 0 0 30 32 0 0 9 11
,
 14 25 0 0 25 27 0 0 0 0 24 1 0 0 40 17
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,34,7,0,0,34,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,7,40],[15,15,0,0,26,15,0,0,0,0,30,9,0,0,32,11],[14,25,0,0,25,27,0,0,0,0,24,40,0,0,1,17] >;

D10.16SD16 in GAP, Magma, Sage, TeX

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

G:=Group("D10.16SD16");
// GroupNames label

G:=SmallGroup(320,404);
// by ID

G=gap.SmallGroup(320,404);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,926,219,851,438,102,12550]);
// Polycyclic

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

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