metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊1SD16, D20.18D4, C42.32D10, C4⋊C8⋊8D5, C4⋊3(C40⋊C2), C4.129(D4×D5), C20⋊2Q8⋊11C2, (C4×D20).10C2, C20.338(C2×D4), (C2×C8).128D10, (C2×C4).131D20, (C2×C20).120D4, C5⋊2(D4.D4), (C4×C20).67C22, C10.10(C2×SD16), C20.327(C4○D4), C20.44D4⋊12C2, C2.10(C4⋊D20), C10.37(C4⋊D4), (C2×C40).138C22, (C2×C20).751C23, C4.43(Q8⋊2D5), C22.114(C2×D20), C2.17(C8.D10), (C2×D20).200C22, C10.14(C8.C22), C4⋊Dic5.272C22, (C2×Dic10).16C22, (C5×C4⋊C8)⋊10C2, (C2×C40⋊C2).5C2, C2.13(C2×C40⋊C2), (C2×C10).134(C2×D4), (C2×C4).696(C22×D5), SmallGroup(320,468)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊SD16
G = < a,b,c | a20=b8=c2=1, bab-1=a11, cac=a9, cbc=b3 >
Subgroups: 566 in 120 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×2], C20, D10 [×4], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C40 [×2], Dic10 [×4], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×3], C2×C20 [×3], C22×D5, D4.D4, C40⋊C2 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, C4×C20, C2×C40 [×2], C2×Dic10 [×2], C2×C4×D5, C2×D20, C20.44D4 [×2], C5×C4⋊C8, C20⋊2Q8, C4×D20, C2×C40⋊C2 [×2], C20⋊SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8.C22, D20 [×2], C22×D5, D4.D4, C40⋊C2 [×2], C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, C2×C40⋊C2, C8.D10, C20⋊SD16
►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)
(1 104 152 66 42 100 28 133)(2 115 153 77 43 91 29 124)(3 106 154 68 44 82 30 135)(4 117 155 79 45 93 31 126)(5 108 156 70 46 84 32 137)(6 119 157 61 47 95 33 128)(7 110 158 72 48 86 34 139)(8 101 159 63 49 97 35 130)(9 112 160 74 50 88 36 121)(10 103 141 65 51 99 37 132)(11 114 142 76 52 90 38 123)(12 105 143 67 53 81 39 134)(13 116 144 78 54 92 40 125)(14 107 145 69 55 83 21 136)(15 118 146 80 56 94 22 127)(16 109 147 71 57 85 23 138)(17 120 148 62 58 96 24 129)(18 111 149 73 59 87 25 140)(19 102 150 64 60 98 26 131)(20 113 151 75 41 89 27 122)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 149)(22 158)(23 147)(24 156)(25 145)(26 154)(27 143)(28 152)(29 141)(30 150)(31 159)(32 148)(33 157)(34 146)(35 155)(36 144)(37 153)(38 142)(39 151)(40 160)(41 53)(43 51)(44 60)(45 49)(46 58)(48 56)(50 54)(55 59)(61 119)(62 108)(63 117)(64 106)(65 115)(66 104)(67 113)(68 102)(69 111)(70 120)(71 109)(72 118)(73 107)(74 116)(75 105)(76 114)(77 103)(78 112)(79 101)(80 110)(81 122)(82 131)(83 140)(84 129)(85 138)(86 127)(87 136)(88 125)(89 134)(90 123)(91 132)(92 121)(93 130)(94 139)(95 128)(96 137)(97 126)(98 135)(99 124)(100 133)
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), (1,104,152,66,42,100,28,133)(2,115,153,77,43,91,29,124)(3,106,154,68,44,82,30,135)(4,117,155,79,45,93,31,126)(5,108,156,70,46,84,32,137)(6,119,157,61,47,95,33,128)(7,110,158,72,48,86,34,139)(8,101,159,63,49,97,35,130)(9,112,160,74,50,88,36,121)(10,103,141,65,51,99,37,132)(11,114,142,76,52,90,38,123)(12,105,143,67,53,81,39,134)(13,116,144,78,54,92,40,125)(14,107,145,69,55,83,21,136)(15,118,146,80,56,94,22,127)(16,109,147,71,57,85,23,138)(17,120,148,62,58,96,24,129)(18,111,149,73,59,87,25,140)(19,102,150,64,60,98,26,131)(20,113,151,75,41,89,27,122), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,149)(22,158)(23,147)(24,156)(25,145)(26,154)(27,143)(28,152)(29,141)(30,150)(31,159)(32,148)(33,157)(34,146)(35,155)(36,144)(37,153)(38,142)(39,151)(40,160)(41,53)(43,51)(44,60)(45,49)(46,58)(48,56)(50,54)(55,59)(61,119)(62,108)(63,117)(64,106)(65,115)(66,104)(67,113)(68,102)(69,111)(70,120)(71,109)(72,118)(73,107)(74,116)(75,105)(76,114)(77,103)(78,112)(79,101)(80,110)(81,122)(82,131)(83,140)(84,129)(85,138)(86,127)(87,136)(88,125)(89,134)(90,123)(91,132)(92,121)(93,130)(94,139)(95,128)(96,137)(97,126)(98,135)(99,124)(100,133)>;
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), (1,104,152,66,42,100,28,133)(2,115,153,77,43,91,29,124)(3,106,154,68,44,82,30,135)(4,117,155,79,45,93,31,126)(5,108,156,70,46,84,32,137)(6,119,157,61,47,95,33,128)(7,110,158,72,48,86,34,139)(8,101,159,63,49,97,35,130)(9,112,160,74,50,88,36,121)(10,103,141,65,51,99,37,132)(11,114,142,76,52,90,38,123)(12,105,143,67,53,81,39,134)(13,116,144,78,54,92,40,125)(14,107,145,69,55,83,21,136)(15,118,146,80,56,94,22,127)(16,109,147,71,57,85,23,138)(17,120,148,62,58,96,24,129)(18,111,149,73,59,87,25,140)(19,102,150,64,60,98,26,131)(20,113,151,75,41,89,27,122), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,149)(22,158)(23,147)(24,156)(25,145)(26,154)(27,143)(28,152)(29,141)(30,150)(31,159)(32,148)(33,157)(34,146)(35,155)(36,144)(37,153)(38,142)(39,151)(40,160)(41,53)(43,51)(44,60)(45,49)(46,58)(48,56)(50,54)(55,59)(61,119)(62,108)(63,117)(64,106)(65,115)(66,104)(67,113)(68,102)(69,111)(70,120)(71,109)(72,118)(73,107)(74,116)(75,105)(76,114)(77,103)(78,112)(79,101)(80,110)(81,122)(82,131)(83,140)(84,129)(85,138)(86,127)(87,136)(88,125)(89,134)(90,123)(91,132)(92,121)(93,130)(94,139)(95,128)(96,137)(97,126)(98,135)(99,124)(100,133) );
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)], [(1,104,152,66,42,100,28,133),(2,115,153,77,43,91,29,124),(3,106,154,68,44,82,30,135),(4,117,155,79,45,93,31,126),(5,108,156,70,46,84,32,137),(6,119,157,61,47,95,33,128),(7,110,158,72,48,86,34,139),(8,101,159,63,49,97,35,130),(9,112,160,74,50,88,36,121),(10,103,141,65,51,99,37,132),(11,114,142,76,52,90,38,123),(12,105,143,67,53,81,39,134),(13,116,144,78,54,92,40,125),(14,107,145,69,55,83,21,136),(15,118,146,80,56,94,22,127),(16,109,147,71,57,85,23,138),(17,120,148,62,58,96,24,129),(18,111,149,73,59,87,25,140),(19,102,150,64,60,98,26,131),(20,113,151,75,41,89,27,122)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,149),(22,158),(23,147),(24,156),(25,145),(26,154),(27,143),(28,152),(29,141),(30,150),(31,159),(32,148),(33,157),(34,146),(35,155),(36,144),(37,153),(38,142),(39,151),(40,160),(41,53),(43,51),(44,60),(45,49),(46,58),(48,56),(50,54),(55,59),(61,119),(62,108),(63,117),(64,106),(65,115),(66,104),(67,113),(68,102),(69,111),(70,120),(71,109),(72,118),(73,107),(74,116),(75,105),(76,114),(77,103),(78,112),(79,101),(80,110),(81,122),(82,131),(83,140),(84,129),(85,138),(86,127),(87,136),(88,125),(89,134),(90,123),(91,132),(92,121),(93,130),(94,139),(95,128),(96,137),(97,126),(98,135),(99,124),(100,133)])
59 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 | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 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 | D4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | D20 | C40⋊C2 | C8.C22 | D4×D5 | Q8⋊2D5 | C8.D10 |
| kernel | C20⋊SD16 | C20.44D4 | C5×C4⋊C8 | C20⋊2Q8 | C4×D20 | C2×C40⋊C2 | D20 | C2×C20 | C4⋊C8 | C20 | C20 | C42 | C2×C8 | C2×C4 | C4 | C10 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of C20⋊SD16 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 24 |
| 0 | 0 | 0 | 0 | 17 | 1 |
| 36 | 19 | 0 | 0 | 0 | 0 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,40,17,0,0,0,0,24,1],[36,34,0,0,0,0,19,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C20⋊SD16 in GAP, Magma, Sage, TeX
C_{20}\rtimes {\rm SD}_{16} % in TeX
G:=Group("C20:SD16"); // GroupNames label
G:=SmallGroup(320,468);
// by ID
G=gap.SmallGroup(320,468);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^11,c*a*c=a^9,c*b*c=b^3>;
// generators/relations