metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊5SD16, (C5×Q8)⋊4D4, (Q8×Dic5)⋊4C2, (C2×SD16)⋊7D5, Q8⋊1(C5⋊D4), C5⋊5(C4⋊SD16), (C2×D4).68D10, C20.171(C2×D4), (C2×C8).144D10, C20⋊D4.7C2, D20⋊5C4⋊33C2, C2.27(D5×SD16), (C10×SD16)⋊18C2, C20.97(C4○D4), D4⋊Dic5⋊32C2, (C2×Q8).113D10, C20.8Q8⋊33C2, (C2×Dic5).75D4, C10.44(C2×SD16), C22.261(D4×D5), C4.10(D4⋊2D5), C2.26(D40⋊C2), C10.75(C8⋊C22), (C2×C40).291C22, (C2×C20).440C23, (D4×C10).89C22, (Q8×C10).70C22, C10.113(C4⋊D4), (C2×D20).121C22, C4⋊Dic5.170C22, (C4×Dic5).53C22, C2.25(Dic5⋊D4), (C2×Q8⋊D5)⋊16C2, C4.39(C2×C5⋊D4), (C2×C10).352(C2×D4), (C2×C4).529(C22×D5), (C2×C5⋊2C8).152C22, SmallGroup(320,790)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊5SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=c3 >
Subgroups: 582 in 128 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C42, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C2×SD16, C5⋊2C8, C40, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, C4⋊SD16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, Q8⋊D5, C2×C40, C5×SD16, C2×D20, C2×C5⋊D4, D4×C10, Q8×C10, C20.8Q8, D20⋊5C4, D4⋊Dic5, C20⋊D4, C2×Q8⋊D5, Q8×Dic5, C10×SD16, Dic5⋊5SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, C4⋊SD16, D4×D5, D4⋊2D5, C2×C5⋊D4, D5×SD16, D40⋊C2, Dic5⋊D4, Dic5⋊5SD16
►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)
(1 108 6 103)(2 107 7 102)(3 106 8 101)(4 105 9 110)(5 104 10 109)(11 56 16 51)(12 55 17 60)(13 54 18 59)(14 53 19 58)(15 52 20 57)(21 95 26 100)(22 94 27 99)(23 93 28 98)(24 92 29 97)(25 91 30 96)(31 115 36 120)(32 114 37 119)(33 113 38 118)(34 112 39 117)(35 111 40 116)(41 125 46 130)(42 124 47 129)(43 123 48 128)(44 122 49 127)(45 121 50 126)(61 155 66 160)(62 154 67 159)(63 153 68 158)(64 152 69 157)(65 151 70 156)(71 145 76 150)(72 144 77 149)(73 143 78 148)(74 142 79 147)(75 141 80 146)(81 135 86 140)(82 134 87 139)(83 133 88 138)(84 132 89 137)(85 131 90 136)
(1 148 38 14 28 138 48 158)(2 149 39 15 29 139 49 159)(3 150 40 16 30 140 50 160)(4 141 31 17 21 131 41 151)(5 142 32 18 22 132 42 152)(6 143 33 19 23 133 43 153)(7 144 34 20 24 134 44 154)(8 145 35 11 25 135 45 155)(9 146 36 12 26 136 46 156)(10 147 37 13 27 137 47 157)(51 91 81 121 61 101 71 111)(52 92 82 122 62 102 72 112)(53 93 83 123 63 103 73 113)(54 94 84 124 64 104 74 114)(55 95 85 125 65 105 75 115)(56 96 86 126 66 106 76 116)(57 97 87 127 67 107 77 117)(58 98 88 128 68 108 78 118)(59 99 89 129 69 109 79 119)(60 100 90 130 70 110 80 120)
(1 98)(2 99)(3 100)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
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), (1,108,6,103)(2,107,7,102)(3,106,8,101)(4,105,9,110)(5,104,10,109)(11,56,16,51)(12,55,17,60)(13,54,18,59)(14,53,19,58)(15,52,20,57)(21,95,26,100)(22,94,27,99)(23,93,28,98)(24,92,29,97)(25,91,30,96)(31,115,36,120)(32,114,37,119)(33,113,38,118)(34,112,39,117)(35,111,40,116)(41,125,46,130)(42,124,47,129)(43,123,48,128)(44,122,49,127)(45,121,50,126)(61,155,66,160)(62,154,67,159)(63,153,68,158)(64,152,69,157)(65,151,70,156)(71,145,76,150)(72,144,77,149)(73,143,78,148)(74,142,79,147)(75,141,80,146)(81,135,86,140)(82,134,87,139)(83,133,88,138)(84,132,89,137)(85,131,90,136), (1,148,38,14,28,138,48,158)(2,149,39,15,29,139,49,159)(3,150,40,16,30,140,50,160)(4,141,31,17,21,131,41,151)(5,142,32,18,22,132,42,152)(6,143,33,19,23,133,43,153)(7,144,34,20,24,134,44,154)(8,145,35,11,25,135,45,155)(9,146,36,12,26,136,46,156)(10,147,37,13,27,137,47,157)(51,91,81,121,61,101,71,111)(52,92,82,122,62,102,72,112)(53,93,83,123,63,103,73,113)(54,94,84,124,64,104,74,114)(55,95,85,125,65,105,75,115)(56,96,86,126,66,106,76,116)(57,97,87,127,67,107,77,117)(58,98,88,128,68,108,78,118)(59,99,89,129,69,109,79,119)(60,100,90,130,70,110,80,120), (1,98)(2,99)(3,100)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)>;
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), (1,108,6,103)(2,107,7,102)(3,106,8,101)(4,105,9,110)(5,104,10,109)(11,56,16,51)(12,55,17,60)(13,54,18,59)(14,53,19,58)(15,52,20,57)(21,95,26,100)(22,94,27,99)(23,93,28,98)(24,92,29,97)(25,91,30,96)(31,115,36,120)(32,114,37,119)(33,113,38,118)(34,112,39,117)(35,111,40,116)(41,125,46,130)(42,124,47,129)(43,123,48,128)(44,122,49,127)(45,121,50,126)(61,155,66,160)(62,154,67,159)(63,153,68,158)(64,152,69,157)(65,151,70,156)(71,145,76,150)(72,144,77,149)(73,143,78,148)(74,142,79,147)(75,141,80,146)(81,135,86,140)(82,134,87,139)(83,133,88,138)(84,132,89,137)(85,131,90,136), (1,148,38,14,28,138,48,158)(2,149,39,15,29,139,49,159)(3,150,40,16,30,140,50,160)(4,141,31,17,21,131,41,151)(5,142,32,18,22,132,42,152)(6,143,33,19,23,133,43,153)(7,144,34,20,24,134,44,154)(8,145,35,11,25,135,45,155)(9,146,36,12,26,136,46,156)(10,147,37,13,27,137,47,157)(51,91,81,121,61,101,71,111)(52,92,82,122,62,102,72,112)(53,93,83,123,63,103,73,113)(54,94,84,124,64,104,74,114)(55,95,85,125,65,105,75,115)(56,96,86,126,66,106,76,116)(57,97,87,127,67,107,77,117)(58,98,88,128,68,108,78,118)(59,99,89,129,69,109,79,119)(60,100,90,130,70,110,80,120), (1,98)(2,99)(3,100)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160) );
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)], [(1,108,6,103),(2,107,7,102),(3,106,8,101),(4,105,9,110),(5,104,10,109),(11,56,16,51),(12,55,17,60),(13,54,18,59),(14,53,19,58),(15,52,20,57),(21,95,26,100),(22,94,27,99),(23,93,28,98),(24,92,29,97),(25,91,30,96),(31,115,36,120),(32,114,37,119),(33,113,38,118),(34,112,39,117),(35,111,40,116),(41,125,46,130),(42,124,47,129),(43,123,48,128),(44,122,49,127),(45,121,50,126),(61,155,66,160),(62,154,67,159),(63,153,68,158),(64,152,69,157),(65,151,70,156),(71,145,76,150),(72,144,77,149),(73,143,78,148),(74,142,79,147),(75,141,80,146),(81,135,86,140),(82,134,87,139),(83,133,88,138),(84,132,89,137),(85,131,90,136)], [(1,148,38,14,28,138,48,158),(2,149,39,15,29,139,49,159),(3,150,40,16,30,140,50,160),(4,141,31,17,21,131,41,151),(5,142,32,18,22,132,42,152),(6,143,33,19,23,133,43,153),(7,144,34,20,24,134,44,154),(8,145,35,11,25,135,45,155),(9,146,36,12,26,136,46,156),(10,147,37,13,27,137,47,157),(51,91,81,121,61,101,71,111),(52,92,82,122,62,102,72,112),(53,93,83,123,63,103,73,113),(54,94,84,124,64,104,74,114),(55,95,85,125,65,105,75,115),(56,96,86,126,66,106,76,116),(57,97,87,127,67,107,77,117),(58,98,88,128,68,108,78,118),(59,99,89,129,69,109,79,119),(60,100,90,130,70,110,80,120)], [(1,98),(2,99),(3,100),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 4 | 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 | 40 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 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 | SD16 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4⋊2D5 | D4×D5 | D5×SD16 | D40⋊C2 |
| kernel | Dic5⋊5SD16 | C20.8Q8 | D20⋊5C4 | D4⋊Dic5 | C20⋊D4 | C2×Q8⋊D5 | Q8×Dic5 | C10×SD16 | C2×Dic5 | C5×Q8 | C2×SD16 | Dic5 | C20 | C2×C8 | C2×D4 | C2×Q8 | Q8 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊5SD16 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 26 | 15 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[24,40,0,0,1,17,0,0,0,0,15,26,0,0,15,15],[24,40,0,0,1,17,0,0,0,0,40,0,0,0,0,1] >;
Dic5⋊5SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_5{\rm SD}_{16} % in TeX
G:=Group("Dic5:5SD16"); // GroupNames label
G:=SmallGroup(320,790);
// by ID
G=gap.SmallGroup(320,790);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^3>;
// generators/relations