metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.16SD16, C40⋊6C4⋊12C2, D4⋊C4⋊11D5, C4⋊C4.136D10, (C2×D4).25D10, (C2×C8).115D10, D4⋊Dic5⋊8C2, C20⋊2D4.3C2, C20.Q8⋊4C2, D10⋊1C8⋊10C2, C2.12(D5×SD16), C4.52(C4○D20), (C2×Dic5).29D4, C10.24(C2×SD16), C22.175(D4×D5), C20.150(C4○D4), C4.79(D4⋊2D5), C2.14(D8⋊D5), C10.32(C8⋊C22), (C2×C20).217C23, (C2×C40).126C22, (C22×D5).110D4, (D4×C10).38C22, C5⋊2(C23.46D4), C4⋊Dic5.71C22, C2.13(D10.12D4), C10.21(C22.D4), (D5×C4⋊C4)⋊4C2, (C5×D4⋊C4)⋊11C2, (C2×C4×D5).14C22, (C2×C10).230(C2×D4), (C5×C4⋊C4).18C22, (C2×C5⋊2C8).16C22, (C2×C4).324(C22×D5), SmallGroup(320,404)
Series: Derived ►Chief ►Lower central ►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, C5⋊2C8, 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×C5⋊2C8, 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, C40⋊6C4, D10⋊1C8, D4⋊Dic5, C5×D4⋊C4, D5×C4⋊C4, C20⋊2D4, 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, D4⋊2D5, D10.12D4, D8⋊D5, D5×SD16, D10.16SD16
(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