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## G = D10⋊2SD16order 320 = 26·5

### 2nd semidirect product of D10 and SD16 acting via SD16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10⋊2SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C2×C8 — D10⋊2SD16
 Lower central C5 — C10 — C2×C20 — D10⋊2SD16
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for D102SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c3 >

Subgroups: 582 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C88D4, C8×D5, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, Q8×C10, C20.Q8, D205C4, C5×Q8⋊C4, C4⋊D20, D5×C2×C8, C2×Q8⋊D5, D103Q8, D102SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, C22×D5, C88D4, C4○D20, D4×D5, D10⋊D4, D5×SD16, Q8.D10, D102SD16

Smallest permutation representation of D102SD16
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)
(1 143)(2 142)(3 141)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 151)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 130)(28 129)(29 128)(30 127)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 60)(41 133)(42 132)(43 131)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 93)(82 92)(83 91)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)
(1 50 15 64 55 78 28 95)(2 49 16 63 56 77 29 94)(3 48 17 62 57 76 30 93)(4 47 18 61 58 75 21 92)(5 46 19 70 59 74 22 91)(6 45 20 69 60 73 23 100)(7 44 11 68 51 72 24 99)(8 43 12 67 52 71 25 98)(9 42 13 66 53 80 26 97)(10 41 14 65 54 79 27 96)(31 109 125 85 149 140 153 113)(32 108 126 84 150 139 154 112)(33 107 127 83 141 138 155 111)(34 106 128 82 142 137 156 120)(35 105 129 81 143 136 157 119)(36 104 130 90 144 135 158 118)(37 103 121 89 145 134 159 117)(38 102 122 88 146 133 160 116)(39 101 123 87 147 132 151 115)(40 110 124 86 148 131 152 114)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 60)(7 59)(8 58)(9 57)(10 56)(11 19)(12 18)(13 17)(14 16)(21 25)(22 24)(26 30)(27 29)(31 144)(32 143)(33 142)(34 141)(35 150)(36 149)(37 148)(38 147)(39 146)(40 145)(41 94)(42 93)(43 92)(44 91)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(61 71)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 139)(82 138)(83 137)(84 136)(85 135)(86 134)(87 133)(88 132)(89 131)(90 140)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 120)(108 119)(109 118)(110 117)(121 124)(122 123)(125 130)(126 129)(127 128)(151 160)(152 159)(153 158)(154 157)(155 156)

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

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

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

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 20A 20B 20C 20D 20E ··· 20L 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 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 40 2 2 8 8 10 10 40 2 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 4 4 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 D5×SD16 Q8.D10 kernel D10⋊2SD16 C20.Q8 D20⋊5C4 C5×Q8⋊C4 C4⋊D20 D5×C2×C8 C2×Q8⋊D5 D10⋊3Q8 C5⋊2C8 C2×Dic5 C22×D5 Q8⋊C4 C20 D10 C4⋊C4 C2×C8 C2×Q8 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 D102SD16 in GL4(𝔽41) generated by

 7 6 0 0 34 0 0 0 0 0 40 0 0 0 0 40
,
 27 2 0 0 5 14 0 0 0 0 0 9 0 0 32 0
,
 3 21 0 0 21 38 0 0 0 0 26 26 0 0 15 26
,
 34 35 0 0 8 7 0 0 0 0 1 0 0 0 0 40
G:=sub<GL(4,GF(41))| [7,34,0,0,6,0,0,0,0,0,40,0,0,0,0,40],[27,5,0,0,2,14,0,0,0,0,0,32,0,0,9,0],[3,21,0,0,21,38,0,0,0,0,26,15,0,0,26,26],[34,8,0,0,35,7,0,0,0,0,1,0,0,0,0,40] >;

D102SD16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("D10:2SD16");
// GroupNames label

G:=SmallGroup(320,434);
// by ID

G=gap.SmallGroup(320,434);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,297,136,438,102,12550]);
// Polycyclic

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

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