metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊7SD16, Q8⋊D5⋊5C4, Q8⋊1(C4×D5), C5⋊4(C4×SD16), (Q8×Dic5)⋊1C2, C10.64(C4×D4), C2.3(D5×SD16), C4⋊C4.142D10, Q8⋊C4⋊21D5, (C8×Dic5)⋊21C2, D20.16(C2×C4), (C2×C8).204D10, (C2×Q8).96D10, C20.Q8⋊9C2, C22.73(D4×D5), D20⋊8C4.1C2, D20⋊5C4.6C2, C10.66(C4○D8), C20.44(C22×C4), C10.26(C2×SD16), C20.154(C4○D4), C2.1(Q8.D10), C4.51(D4⋊2D5), (C2×C40).189C22, (C2×C20).228C23, (C2×Dic5).270D4, (C2×D20).60C22, C4⋊Dic5.78C22, (Q8×C10).11C22, C2.18(Dic5⋊4D4), (C4×Dic5).255C22, C4.9(C2×C4×D5), C5⋊2C8⋊19(C2×C4), (C5×Q8)⋊10(C2×C4), (C2×Q8⋊D5).1C2, (C5×Q8⋊C4)⋊15C2, (C2×C10).241(C2×D4), (C5×C4⋊C4).29C22, (C2×C4).335(C22×D5), (C2×C5⋊2C8).222C22, SmallGroup(320,415)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊7SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c3 >
Subgroups: 486 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C4×SD16, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, Q8⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, Q8×C10, C20.Q8, C8×Dic5, D20⋊5C4, C5×Q8⋊C4, D20⋊8C4, C2×Q8⋊D5, Q8×Dic5, Dic5⋊7SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, SD16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×SD16, C4○D8, C4×D5, C22×D5, C4×SD16, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, D5×SD16, Q8.D10, Dic5⋊7SD16
►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 58 16 53)(12 57 17 52)(13 56 18 51)(14 55 19 60)(15 54 20 59)(21 97 26 92)(22 96 27 91)(23 95 28 100)(24 94 29 99)(25 93 30 98)(31 119 36 114)(32 118 37 113)(33 117 38 112)(34 116 39 111)(35 115 40 120)(41 125 46 130)(42 124 47 129)(43 123 48 128)(44 122 49 127)(45 121 50 126)(61 157 66 152)(62 156 67 151)(63 155 68 160)(64 154 69 159)(65 153 70 158)(71 147 76 142)(72 146 77 141)(73 145 78 150)(74 144 79 149)(75 143 80 148)(81 137 86 132)(82 136 87 131)(83 135 88 140)(84 134 89 139)(85 133 90 138)
(1 80 40 60 28 88 48 68)(2 79 31 59 29 87 49 67)(3 78 32 58 30 86 50 66)(4 77 33 57 21 85 41 65)(5 76 34 56 22 84 42 64)(6 75 35 55 23 83 43 63)(7 74 36 54 24 82 44 62)(8 73 37 53 25 81 45 61)(9 72 38 52 26 90 46 70)(10 71 39 51 27 89 47 69)(11 93 137 121 157 101 145 113)(12 92 138 130 158 110 146 112)(13 91 139 129 159 109 147 111)(14 100 140 128 160 108 148 120)(15 99 131 127 151 107 149 119)(16 98 132 126 152 106 150 118)(17 97 133 125 153 105 141 117)(18 96 134 124 154 104 142 116)(19 95 135 123 155 103 143 115)(20 94 136 122 156 102 144 114)
(1 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 30)(10 29)(11 133)(12 132)(13 131)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(31 39)(32 38)(33 37)(34 36)(41 45)(42 44)(46 50)(47 49)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 90)(59 89)(60 88)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 80)(69 79)(70 78)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 110)(99 109)(100 108)(111 119)(112 118)(113 117)(114 116)(121 125)(122 124)(126 130)(127 129)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 160)(149 159)(150 158)
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)(11,58,16,53)(12,57,17,52)(13,56,18,51)(14,55,19,60)(15,54,20,59)(21,97,26,92)(22,96,27,91)(23,95,28,100)(24,94,29,99)(25,93,30,98)(31,119,36,114)(32,118,37,113)(33,117,38,112)(34,116,39,111)(35,115,40,120)(41,125,46,130)(42,124,47,129)(43,123,48,128)(44,122,49,127)(45,121,50,126)(61,157,66,152)(62,156,67,151)(63,155,68,160)(64,154,69,159)(65,153,70,158)(71,147,76,142)(72,146,77,141)(73,145,78,150)(74,144,79,149)(75,143,80,148)(81,137,86,132)(82,136,87,131)(83,135,88,140)(84,134,89,139)(85,133,90,138), (1,80,40,60,28,88,48,68)(2,79,31,59,29,87,49,67)(3,78,32,58,30,86,50,66)(4,77,33,57,21,85,41,65)(5,76,34,56,22,84,42,64)(6,75,35,55,23,83,43,63)(7,74,36,54,24,82,44,62)(8,73,37,53,25,81,45,61)(9,72,38,52,26,90,46,70)(10,71,39,51,27,89,47,69)(11,93,137,121,157,101,145,113)(12,92,138,130,158,110,146,112)(13,91,139,129,159,109,147,111)(14,100,140,128,160,108,148,120)(15,99,131,127,151,107,149,119)(16,98,132,126,152,106,150,118)(17,97,133,125,153,105,141,117)(18,96,134,124,154,104,142,116)(19,95,135,123,155,103,143,115)(20,94,136,122,156,102,144,114), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,133)(12,132)(13,131)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(31,39)(32,38)(33,37)(34,36)(41,45)(42,44)(46,50)(47,49)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,90)(59,89)(60,88)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,80)(69,79)(70,78)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,110)(99,109)(100,108)(111,119)(112,118)(113,117)(114,116)(121,125)(122,124)(126,130)(127,129)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,160)(149,159)(150,158)>;
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)(11,58,16,53)(12,57,17,52)(13,56,18,51)(14,55,19,60)(15,54,20,59)(21,97,26,92)(22,96,27,91)(23,95,28,100)(24,94,29,99)(25,93,30,98)(31,119,36,114)(32,118,37,113)(33,117,38,112)(34,116,39,111)(35,115,40,120)(41,125,46,130)(42,124,47,129)(43,123,48,128)(44,122,49,127)(45,121,50,126)(61,157,66,152)(62,156,67,151)(63,155,68,160)(64,154,69,159)(65,153,70,158)(71,147,76,142)(72,146,77,141)(73,145,78,150)(74,144,79,149)(75,143,80,148)(81,137,86,132)(82,136,87,131)(83,135,88,140)(84,134,89,139)(85,133,90,138), (1,80,40,60,28,88,48,68)(2,79,31,59,29,87,49,67)(3,78,32,58,30,86,50,66)(4,77,33,57,21,85,41,65)(5,76,34,56,22,84,42,64)(6,75,35,55,23,83,43,63)(7,74,36,54,24,82,44,62)(8,73,37,53,25,81,45,61)(9,72,38,52,26,90,46,70)(10,71,39,51,27,89,47,69)(11,93,137,121,157,101,145,113)(12,92,138,130,158,110,146,112)(13,91,139,129,159,109,147,111)(14,100,140,128,160,108,148,120)(15,99,131,127,151,107,149,119)(16,98,132,126,152,106,150,118)(17,97,133,125,153,105,141,117)(18,96,134,124,154,104,142,116)(19,95,135,123,155,103,143,115)(20,94,136,122,156,102,144,114), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,133)(12,132)(13,131)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(31,39)(32,38)(33,37)(34,36)(41,45)(42,44)(46,50)(47,49)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,90)(59,89)(60,88)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,80)(69,79)(70,78)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,110)(99,109)(100,108)(111,119)(112,118)(113,117)(114,116)(121,125)(122,124)(126,130)(127,129)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,160)(149,159)(150,158) );
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),(11,58,16,53),(12,57,17,52),(13,56,18,51),(14,55,19,60),(15,54,20,59),(21,97,26,92),(22,96,27,91),(23,95,28,100),(24,94,29,99),(25,93,30,98),(31,119,36,114),(32,118,37,113),(33,117,38,112),(34,116,39,111),(35,115,40,120),(41,125,46,130),(42,124,47,129),(43,123,48,128),(44,122,49,127),(45,121,50,126),(61,157,66,152),(62,156,67,151),(63,155,68,160),(64,154,69,159),(65,153,70,158),(71,147,76,142),(72,146,77,141),(73,145,78,150),(74,144,79,149),(75,143,80,148),(81,137,86,132),(82,136,87,131),(83,135,88,140),(84,134,89,139),(85,133,90,138)], [(1,80,40,60,28,88,48,68),(2,79,31,59,29,87,49,67),(3,78,32,58,30,86,50,66),(4,77,33,57,21,85,41,65),(5,76,34,56,22,84,42,64),(6,75,35,55,23,83,43,63),(7,74,36,54,24,82,44,62),(8,73,37,53,25,81,45,61),(9,72,38,52,26,90,46,70),(10,71,39,51,27,89,47,69),(11,93,137,121,157,101,145,113),(12,92,138,130,158,110,146,112),(13,91,139,129,159,109,147,111),(14,100,140,128,160,108,148,120),(15,99,131,127,151,107,149,119),(16,98,132,126,152,106,150,118),(17,97,133,125,153,105,141,117),(18,96,134,124,154,104,142,116),(19,95,135,123,155,103,143,115),(20,94,136,122,156,102,144,114)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,30),(10,29),(11,133),(12,132),(13,131),(14,140),(15,139),(16,138),(17,137),(18,136),(19,135),(20,134),(31,39),(32,38),(33,37),(34,36),(41,45),(42,44),(46,50),(47,49),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,90),(59,89),(60,88),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,80),(69,79),(70,78),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,110),(99,109),(100,108),(111,119),(112,118),(113,117),(114,116),(121,125),(122,124),(126,130),(127,129),(141,157),(142,156),(143,155),(144,154),(145,153),(146,152),(147,151),(148,160),(149,159),(150,158)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4×D5 | D4⋊2D5 | D4×D5 | D5×SD16 | Q8.D10 |
| kernel | Dic5⋊7SD16 | C20.Q8 | C8×Dic5 | D20⋊5C4 | C5×Q8⋊C4 | D20⋊8C4 | C2×Q8⋊D5 | Q8×Dic5 | Q8⋊D5 | C2×Dic5 | Q8⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Dic5⋊7SD16 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 9 | 0 | 0 |
| 19 | 19 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 7 | 1 | 0 | 0 |
| 34 | 34 | 0 | 0 |
| 0 | 0 | 11 | 30 |
| 0 | 0 | 26 | 0 |
| 34 | 40 | 0 | 0 |
| 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 40 | 1 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[22,19,0,0,9,19,0,0,0,0,40,0,0,0,0,40],[7,34,0,0,1,34,0,0,0,0,11,26,0,0,30,0],[34,7,0,0,40,7,0,0,0,0,40,40,0,0,0,1] >;
Dic5⋊7SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_7{\rm SD}_{16} % in TeX
G:=Group("Dic5:7SD16"); // GroupNames label
G:=SmallGroup(320,415);
// by ID
G=gap.SmallGroup(320,415);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,120,135,184,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations