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, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4.D4, C40⋊C2, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, C5×C4⋊C8, C20⋊2Q8, C4×D20, C2×C40⋊C2, C20⋊SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8.C22, D20, C22×D5, D4.D4, C40⋊C2, 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 61 46 137 93 117 154 32)(2 72 47 128 94 108 155 23)(3 63 48 139 95 119 156 34)(4 74 49 130 96 110 157 25)(5 65 50 121 97 101 158 36)(6 76 51 132 98 112 159 27)(7 67 52 123 99 103 160 38)(8 78 53 134 100 114 141 29)(9 69 54 125 81 105 142 40)(10 80 55 136 82 116 143 31)(11 71 56 127 83 107 144 22)(12 62 57 138 84 118 145 33)(13 73 58 129 85 109 146 24)(14 64 59 140 86 120 147 35)(15 75 60 131 87 111 148 26)(16 66 41 122 88 102 149 37)(17 77 42 133 89 113 150 28)(18 68 43 124 90 104 151 39)(19 79 44 135 91 115 152 30)(20 70 45 126 92 106 153 21)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 118)(22 107)(23 116)(24 105)(25 114)(26 103)(27 112)(28 101)(29 110)(30 119)(31 108)(32 117)(33 106)(34 115)(35 104)(36 113)(37 102)(38 111)(39 120)(40 109)(41 149)(42 158)(43 147)(44 156)(45 145)(46 154)(47 143)(48 152)(49 141)(50 150)(51 159)(52 148)(53 157)(54 146)(55 155)(56 144)(57 153)(58 142)(59 151)(60 160)(61 137)(62 126)(63 135)(64 124)(65 133)(66 122)(67 131)(68 140)(69 129)(70 138)(71 127)(72 136)(73 125)(74 134)(75 123)(76 132)(77 121)(78 130)(79 139)(80 128)(81 85)(82 94)(84 92)(86 90)(87 99)(89 97)(91 95)(96 100)
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), (1,61,46,137,93,117,154,32)(2,72,47,128,94,108,155,23)(3,63,48,139,95,119,156,34)(4,74,49,130,96,110,157,25)(5,65,50,121,97,101,158,36)(6,76,51,132,98,112,159,27)(7,67,52,123,99,103,160,38)(8,78,53,134,100,114,141,29)(9,69,54,125,81,105,142,40)(10,80,55,136,82,116,143,31)(11,71,56,127,83,107,144,22)(12,62,57,138,84,118,145,33)(13,73,58,129,85,109,146,24)(14,64,59,140,86,120,147,35)(15,75,60,131,87,111,148,26)(16,66,41,122,88,102,149,37)(17,77,42,133,89,113,150,28)(18,68,43,124,90,104,151,39)(19,79,44,135,91,115,152,30)(20,70,45,126,92,106,153,21), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,118)(22,107)(23,116)(24,105)(25,114)(26,103)(27,112)(28,101)(29,110)(30,119)(31,108)(32,117)(33,106)(34,115)(35,104)(36,113)(37,102)(38,111)(39,120)(40,109)(41,149)(42,158)(43,147)(44,156)(45,145)(46,154)(47,143)(48,152)(49,141)(50,150)(51,159)(52,148)(53,157)(54,146)(55,155)(56,144)(57,153)(58,142)(59,151)(60,160)(61,137)(62,126)(63,135)(64,124)(65,133)(66,122)(67,131)(68,140)(69,129)(70,138)(71,127)(72,136)(73,125)(74,134)(75,123)(76,132)(77,121)(78,130)(79,139)(80,128)(81,85)(82,94)(84,92)(86,90)(87,99)(89,97)(91,95)(96,100)>;
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), (1,61,46,137,93,117,154,32)(2,72,47,128,94,108,155,23)(3,63,48,139,95,119,156,34)(4,74,49,130,96,110,157,25)(5,65,50,121,97,101,158,36)(6,76,51,132,98,112,159,27)(7,67,52,123,99,103,160,38)(8,78,53,134,100,114,141,29)(9,69,54,125,81,105,142,40)(10,80,55,136,82,116,143,31)(11,71,56,127,83,107,144,22)(12,62,57,138,84,118,145,33)(13,73,58,129,85,109,146,24)(14,64,59,140,86,120,147,35)(15,75,60,131,87,111,148,26)(16,66,41,122,88,102,149,37)(17,77,42,133,89,113,150,28)(18,68,43,124,90,104,151,39)(19,79,44,135,91,115,152,30)(20,70,45,126,92,106,153,21), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,118)(22,107)(23,116)(24,105)(25,114)(26,103)(27,112)(28,101)(29,110)(30,119)(31,108)(32,117)(33,106)(34,115)(35,104)(36,113)(37,102)(38,111)(39,120)(40,109)(41,149)(42,158)(43,147)(44,156)(45,145)(46,154)(47,143)(48,152)(49,141)(50,150)(51,159)(52,148)(53,157)(54,146)(55,155)(56,144)(57,153)(58,142)(59,151)(60,160)(61,137)(62,126)(63,135)(64,124)(65,133)(66,122)(67,131)(68,140)(69,129)(70,138)(71,127)(72,136)(73,125)(74,134)(75,123)(76,132)(77,121)(78,130)(79,139)(80,128)(81,85)(82,94)(84,92)(86,90)(87,99)(89,97)(91,95)(96,100) );
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)], [(1,61,46,137,93,117,154,32),(2,72,47,128,94,108,155,23),(3,63,48,139,95,119,156,34),(4,74,49,130,96,110,157,25),(5,65,50,121,97,101,158,36),(6,76,51,132,98,112,159,27),(7,67,52,123,99,103,160,38),(8,78,53,134,100,114,141,29),(9,69,54,125,81,105,142,40),(10,80,55,136,82,116,143,31),(11,71,56,127,83,107,144,22),(12,62,57,138,84,118,145,33),(13,73,58,129,85,109,146,24),(14,64,59,140,86,120,147,35),(15,75,60,131,87,111,148,26),(16,66,41,122,88,102,149,37),(17,77,42,133,89,113,150,28),(18,68,43,124,90,104,151,39),(19,79,44,135,91,115,152,30),(20,70,45,126,92,106,153,21)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,118),(22,107),(23,116),(24,105),(25,114),(26,103),(27,112),(28,101),(29,110),(30,119),(31,108),(32,117),(33,106),(34,115),(35,104),(36,113),(37,102),(38,111),(39,120),(40,109),(41,149),(42,158),(43,147),(44,156),(45,145),(46,154),(47,143),(48,152),(49,141),(50,150),(51,159),(52,148),(53,157),(54,146),(55,155),(56,144),(57,153),(58,142),(59,151),(60,160),(61,137),(62,126),(63,135),(64,124),(65,133),(66,122),(67,131),(68,140),(69,129),(70,138),(71,127),(72,136),(73,125),(74,134),(75,123),(76,132),(77,121),(78,130),(79,139),(80,128),(81,85),(82,94),(84,92),(86,90),(87,99),(89,97),(91,95),(96,100)]])
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