metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.17SD16, C4.Q8⋊7D5, C4⋊C4.37D10, (C2×C8).138D10, C4⋊D20.5C2, D20⋊5C4⋊31C2, D20⋊6C4⋊14C2, D10⋊1C8⋊30C2, C2.23(D5×SD16), C20.28(C4○D4), C4.73(C4○D20), C20.Q8⋊17C2, (C2×Dic5).49D4, C10.39(C2×SD16), C22.215(D4×D5), C2.21(D40⋊C2), C10.69(C8⋊C22), (C2×C40).285C22, (C2×C20).279C23, C4.25(Q8⋊2D5), (C2×D20).77C22, (C22×D5).120D4, C5⋊3(C23.46D4), C4⋊Dic5.111C22, C2.12(D10.13D4), C10.42(C22.D4), (D5×C4⋊C4)⋊6C2, (C5×C4.Q8)⋊16C2, (C2×C4×D5).36C22, (C2×C10).284(C2×D4), (C5×C4⋊C4).72C22, (C2×C5⋊2C8).57C22, (C2×C4).382(C22×D5), SmallGroup(320,490)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.17SD16
G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c3 >
Subgroups: 574 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C4.Q8, C2×C4⋊C4, C4⋊D4, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C23.46D4, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C20.Q8, D20⋊6C4, D10⋊1C8, D20⋊5C4, C5×C4.Q8, D5×C4⋊C4, C4⋊D20, D10.17SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C22.D4, C2×SD16, C8⋊C22, C22×D5, C23.46D4, C4○D20, D4×D5, Q8⋊2D5, D10.13D4, D5×SD16, D40⋊C2, D10.17SD16
►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)
(1 147)(2 146)(3 145)(4 144)(5 143)(6 142)(7 141)(8 150)(9 149)(10 148)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 132)(18 131)(19 140)(20 139)(21 44)(22 43)(23 42)(24 41)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 152)(32 151)(33 160)(34 159)(35 158)(36 157)(37 156)(38 155)(39 154)(40 153)(51 128)(52 127)(53 126)(54 125)(55 124)(56 123)(57 122)(58 121)(59 130)(60 129)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 110)(71 119)(72 118)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 120)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 100)(90 99)
(1 63 45 55 31 78 20 90)(2 64 46 56 32 79 11 81)(3 65 47 57 33 80 12 82)(4 66 48 58 34 71 13 83)(5 67 49 59 35 72 14 84)(6 68 50 60 36 73 15 85)(7 69 41 51 37 74 16 86)(8 70 42 52 38 75 17 87)(9 61 43 53 39 76 18 88)(10 62 44 54 40 77 19 89)(21 130 153 118 140 95 148 103)(22 121 154 119 131 96 149 104)(23 122 155 120 132 97 150 105)(24 123 156 111 133 98 141 106)(25 124 157 112 134 99 142 107)(26 125 158 113 135 100 143 108)(27 126 159 114 136 91 144 109)(28 127 160 115 137 92 145 110)(29 128 151 116 138 93 146 101)(30 129 152 117 139 94 147 102)
(1 135 6 140)(2 136 7 131)(3 137 8 132)(4 138 9 133)(5 139 10 134)(11 144 16 149)(12 145 17 150)(13 146 18 141)(14 147 19 142)(15 148 20 143)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 154 46 159)(42 155 47 160)(43 156 48 151)(44 157 49 152)(45 158 50 153)(51 96 56 91)(52 97 57 92)(53 98 58 93)(54 99 59 94)(55 100 60 95)(61 106 66 101)(62 107 67 102)(63 108 68 103)(64 109 69 104)(65 110 70 105)(71 116 76 111)(72 117 77 112)(73 118 78 113)(74 119 79 114)(75 120 80 115)(81 126 86 121)(82 127 87 122)(83 128 88 123)(84 129 89 124)(85 130 90 125)
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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,150)(9,149)(10,148)(11,138)(12,137)(13,136)(14,135)(15,134)(16,133)(17,132)(18,131)(19,140)(20,139)(21,44)(22,43)(23,42)(24,41)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,152)(32,151)(33,160)(34,159)(35,158)(36,157)(37,156)(38,155)(39,154)(40,153)(51,128)(52,127)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,130)(60,129)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,110)(71,119)(72,118)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,120)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,100)(90,99), (1,63,45,55,31,78,20,90)(2,64,46,56,32,79,11,81)(3,65,47,57,33,80,12,82)(4,66,48,58,34,71,13,83)(5,67,49,59,35,72,14,84)(6,68,50,60,36,73,15,85)(7,69,41,51,37,74,16,86)(8,70,42,52,38,75,17,87)(9,61,43,53,39,76,18,88)(10,62,44,54,40,77,19,89)(21,130,153,118,140,95,148,103)(22,121,154,119,131,96,149,104)(23,122,155,120,132,97,150,105)(24,123,156,111,133,98,141,106)(25,124,157,112,134,99,142,107)(26,125,158,113,135,100,143,108)(27,126,159,114,136,91,144,109)(28,127,160,115,137,92,145,110)(29,128,151,116,138,93,146,101)(30,129,152,117,139,94,147,102), (1,135,6,140)(2,136,7,131)(3,137,8,132)(4,138,9,133)(5,139,10,134)(11,144,16,149)(12,145,17,150)(13,146,18,141)(14,147,19,142)(15,148,20,143)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,154,46,159)(42,155,47,160)(43,156,48,151)(44,157,49,152)(45,158,50,153)(51,96,56,91)(52,97,57,92)(53,98,58,93)(54,99,59,94)(55,100,60,95)(61,106,66,101)(62,107,67,102)(63,108,68,103)(64,109,69,104)(65,110,70,105)(71,116,76,111)(72,117,77,112)(73,118,78,113)(74,119,79,114)(75,120,80,115)(81,126,86,121)(82,127,87,122)(83,128,88,123)(84,129,89,124)(85,130,90,125)>;
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), (1,147)(2,146)(3,145)(4,144)(5,143)(6,142)(7,141)(8,150)(9,149)(10,148)(11,138)(12,137)(13,136)(14,135)(15,134)(16,133)(17,132)(18,131)(19,140)(20,139)(21,44)(22,43)(23,42)(24,41)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,152)(32,151)(33,160)(34,159)(35,158)(36,157)(37,156)(38,155)(39,154)(40,153)(51,128)(52,127)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,130)(60,129)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,110)(71,119)(72,118)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,120)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,100)(90,99), (1,63,45,55,31,78,20,90)(2,64,46,56,32,79,11,81)(3,65,47,57,33,80,12,82)(4,66,48,58,34,71,13,83)(5,67,49,59,35,72,14,84)(6,68,50,60,36,73,15,85)(7,69,41,51,37,74,16,86)(8,70,42,52,38,75,17,87)(9,61,43,53,39,76,18,88)(10,62,44,54,40,77,19,89)(21,130,153,118,140,95,148,103)(22,121,154,119,131,96,149,104)(23,122,155,120,132,97,150,105)(24,123,156,111,133,98,141,106)(25,124,157,112,134,99,142,107)(26,125,158,113,135,100,143,108)(27,126,159,114,136,91,144,109)(28,127,160,115,137,92,145,110)(29,128,151,116,138,93,146,101)(30,129,152,117,139,94,147,102), (1,135,6,140)(2,136,7,131)(3,137,8,132)(4,138,9,133)(5,139,10,134)(11,144,16,149)(12,145,17,150)(13,146,18,141)(14,147,19,142)(15,148,20,143)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,154,46,159)(42,155,47,160)(43,156,48,151)(44,157,49,152)(45,158,50,153)(51,96,56,91)(52,97,57,92)(53,98,58,93)(54,99,59,94)(55,100,60,95)(61,106,66,101)(62,107,67,102)(63,108,68,103)(64,109,69,104)(65,110,70,105)(71,116,76,111)(72,117,77,112)(73,118,78,113)(74,119,79,114)(75,120,80,115)(81,126,86,121)(82,127,87,122)(83,128,88,123)(84,129,89,124)(85,130,90,125) );
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)], [(1,147),(2,146),(3,145),(4,144),(5,143),(6,142),(7,141),(8,150),(9,149),(10,148),(11,138),(12,137),(13,136),(14,135),(15,134),(16,133),(17,132),(18,131),(19,140),(20,139),(21,44),(22,43),(23,42),(24,41),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,152),(32,151),(33,160),(34,159),(35,158),(36,157),(37,156),(38,155),(39,154),(40,153),(51,128),(52,127),(53,126),(54,125),(55,124),(56,123),(57,122),(58,121),(59,130),(60,129),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,110),(71,119),(72,118),(73,117),(74,116),(75,115),(76,114),(77,113),(78,112),(79,111),(80,120),(81,98),(82,97),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,100),(90,99)], [(1,63,45,55,31,78,20,90),(2,64,46,56,32,79,11,81),(3,65,47,57,33,80,12,82),(4,66,48,58,34,71,13,83),(5,67,49,59,35,72,14,84),(6,68,50,60,36,73,15,85),(7,69,41,51,37,74,16,86),(8,70,42,52,38,75,17,87),(9,61,43,53,39,76,18,88),(10,62,44,54,40,77,19,89),(21,130,153,118,140,95,148,103),(22,121,154,119,131,96,149,104),(23,122,155,120,132,97,150,105),(24,123,156,111,133,98,141,106),(25,124,157,112,134,99,142,107),(26,125,158,113,135,100,143,108),(27,126,159,114,136,91,144,109),(28,127,160,115,137,92,145,110),(29,128,151,116,138,93,146,101),(30,129,152,117,139,94,147,102)], [(1,135,6,140),(2,136,7,131),(3,137,8,132),(4,138,9,133),(5,139,10,134),(11,144,16,149),(12,145,17,150),(13,146,18,141),(14,147,19,142),(15,148,20,143),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,154,46,159),(42,155,47,160),(43,156,48,151),(44,157,49,152),(45,158,50,153),(51,96,56,91),(52,97,57,92),(53,98,58,93),(54,99,59,94),(55,100,60,95),(61,106,66,101),(62,107,67,102),(63,108,68,103),(64,109,69,104),(65,110,70,105),(71,116,76,111),(72,117,77,112),(73,118,78,113),(74,119,79,114),(75,120,80,115),(81,126,86,121),(82,127,87,122),(83,128,88,123),(84,129,89,124),(85,130,90,125)]])
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 | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 40 | 2 | 2 | 4 | 4 | 8 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4○D20 | C8⋊C22 | Q8⋊2D5 | D4×D5 | D5×SD16 | D40⋊C2 |
| kernel | D10.17SD16 | C20.Q8 | D20⋊6C4 | D10⋊1C8 | D20⋊5C4 | C5×C4.Q8 | D5×C4⋊C4 | C4⋊D20 | C2×Dic5 | C22×D5 | C4.Q8 | C20 | D10 | C4⋊C4 | C2×C8 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10.17SD16 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 34 |
| 0 | 0 | 6 | 35 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 27 |
| 0 | 0 | 27 | 16 |
| 26 | 26 | 0 | 0 |
| 15 | 26 | 0 | 0 |
| 0 | 0 | 18 | 40 |
| 0 | 0 | 36 | 23 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 |
| 0 | 0 | 4 | 39 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,6,0,0,34,35],[1,0,0,0,0,1,0,0,0,0,25,27,0,0,27,16],[26,15,0,0,26,26,0,0,0,0,18,36,0,0,40,23],[0,40,0,0,40,0,0,0,0,0,2,4,0,0,9,39] >;
D10.17SD16 in GAP, Magma, Sage, TeX
D_{10}._{17}{\rm SD}_{16} % in TeX
G:=Group("D10.17SD16"); // GroupNames label
G:=SmallGroup(320,490);
// by ID
G=gap.SmallGroup(320,490);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,926,219,100,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^3>;
// generators/relations