metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊5SD16, D20.24D4, C42.78D10, C4⋊Q8⋊1D5, C4⋊2(Q8⋊D5), C4.56(D4×D5), C20⋊3C8⋊33C2, C20.36(C2×D4), (C4×D20).17C2, (C2×C20).154D4, C5⋊4(D4.D4), (C2×Q8).44D10, C20.80(C4○D4), C4.5(D4⋊2D5), Q8⋊Dic5⋊23C2, C10.75(C2×SD16), C2.14(C20⋊2D4), (C2×C20).401C23, (C4×C20).130C22, (Q8×C10).62C22, C10.105(C4⋊D4), (C2×D20).255C22, C10.95(C8.C22), C4⋊Dic5.346C22, C2.16(C20.C23), (C5×C4⋊Q8)⋊1C2, (C2×Q8⋊D5).6C2, C2.13(C2×Q8⋊D5), (C2×C10).532(C2×D4), (C2×C4).187(C5⋊D4), (C2×C4).498(C22×D5), C22.204(C2×C5⋊D4), (C2×C5⋊2C8).135C22, SmallGroup(320,710)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊5SD16
G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a9, cbc=b3 >
Subgroups: 486 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, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4.D4, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, Q8⋊D5, C4×C20, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C20⋊3C8, Q8⋊Dic5, C4×D20, C2×Q8⋊D5, C5×C4⋊Q8, C20⋊5SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8.C22, C5⋊D4, C22×D5, D4.D4, Q8⋊D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C20⋊2D4, C2×Q8⋊D5, C20.C23, C20⋊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)
(1 109 36 41 86 71 150 132)(2 108 37 60 87 70 151 131)(3 107 38 59 88 69 152 130)(4 106 39 58 89 68 153 129)(5 105 40 57 90 67 154 128)(6 104 21 56 91 66 155 127)(7 103 22 55 92 65 156 126)(8 102 23 54 93 64 157 125)(9 101 24 53 94 63 158 124)(10 120 25 52 95 62 159 123)(11 119 26 51 96 61 160 122)(12 118 27 50 97 80 141 121)(13 117 28 49 98 79 142 140)(14 116 29 48 99 78 143 139)(15 115 30 47 100 77 144 138)(16 114 31 46 81 76 145 137)(17 113 32 45 82 75 146 136)(18 112 33 44 83 74 147 135)(19 111 34 43 84 73 148 134)(20 110 35 42 85 72 149 133)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 155)(22 144)(23 153)(24 142)(25 151)(26 160)(27 149)(28 158)(29 147)(30 156)(31 145)(32 154)(33 143)(34 152)(35 141)(36 150)(37 159)(38 148)(39 157)(40 146)(41 109)(42 118)(43 107)(44 116)(45 105)(46 114)(47 103)(48 112)(49 101)(50 110)(51 119)(52 108)(53 117)(54 106)(55 115)(56 104)(57 113)(58 102)(59 111)(60 120)(61 122)(62 131)(63 140)(64 129)(65 138)(66 127)(67 136)(68 125)(69 134)(70 123)(71 132)(72 121)(73 130)(74 139)(75 128)(76 137)(77 126)(78 135)(79 124)(80 133)(82 90)(83 99)(84 88)(85 97)(87 95)(89 93)(92 100)(94 98)
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), (1,109,36,41,86,71,150,132)(2,108,37,60,87,70,151,131)(3,107,38,59,88,69,152,130)(4,106,39,58,89,68,153,129)(5,105,40,57,90,67,154,128)(6,104,21,56,91,66,155,127)(7,103,22,55,92,65,156,126)(8,102,23,54,93,64,157,125)(9,101,24,53,94,63,158,124)(10,120,25,52,95,62,159,123)(11,119,26,51,96,61,160,122)(12,118,27,50,97,80,141,121)(13,117,28,49,98,79,142,140)(14,116,29,48,99,78,143,139)(15,115,30,47,100,77,144,138)(16,114,31,46,81,76,145,137)(17,113,32,45,82,75,146,136)(18,112,33,44,83,74,147,135)(19,111,34,43,84,73,148,134)(20,110,35,42,85,72,149,133), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,155)(22,144)(23,153)(24,142)(25,151)(26,160)(27,149)(28,158)(29,147)(30,156)(31,145)(32,154)(33,143)(34,152)(35,141)(36,150)(37,159)(38,148)(39,157)(40,146)(41,109)(42,118)(43,107)(44,116)(45,105)(46,114)(47,103)(48,112)(49,101)(50,110)(51,119)(52,108)(53,117)(54,106)(55,115)(56,104)(57,113)(58,102)(59,111)(60,120)(61,122)(62,131)(63,140)(64,129)(65,138)(66,127)(67,136)(68,125)(69,134)(70,123)(71,132)(72,121)(73,130)(74,139)(75,128)(76,137)(77,126)(78,135)(79,124)(80,133)(82,90)(83,99)(84,88)(85,97)(87,95)(89,93)(92,100)(94,98)>;
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), (1,109,36,41,86,71,150,132)(2,108,37,60,87,70,151,131)(3,107,38,59,88,69,152,130)(4,106,39,58,89,68,153,129)(5,105,40,57,90,67,154,128)(6,104,21,56,91,66,155,127)(7,103,22,55,92,65,156,126)(8,102,23,54,93,64,157,125)(9,101,24,53,94,63,158,124)(10,120,25,52,95,62,159,123)(11,119,26,51,96,61,160,122)(12,118,27,50,97,80,141,121)(13,117,28,49,98,79,142,140)(14,116,29,48,99,78,143,139)(15,115,30,47,100,77,144,138)(16,114,31,46,81,76,145,137)(17,113,32,45,82,75,146,136)(18,112,33,44,83,74,147,135)(19,111,34,43,84,73,148,134)(20,110,35,42,85,72,149,133), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,155)(22,144)(23,153)(24,142)(25,151)(26,160)(27,149)(28,158)(29,147)(30,156)(31,145)(32,154)(33,143)(34,152)(35,141)(36,150)(37,159)(38,148)(39,157)(40,146)(41,109)(42,118)(43,107)(44,116)(45,105)(46,114)(47,103)(48,112)(49,101)(50,110)(51,119)(52,108)(53,117)(54,106)(55,115)(56,104)(57,113)(58,102)(59,111)(60,120)(61,122)(62,131)(63,140)(64,129)(65,138)(66,127)(67,136)(68,125)(69,134)(70,123)(71,132)(72,121)(73,130)(74,139)(75,128)(76,137)(77,126)(78,135)(79,124)(80,133)(82,90)(83,99)(84,88)(85,97)(87,95)(89,93)(92,100)(94,98) );
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)], [(1,109,36,41,86,71,150,132),(2,108,37,60,87,70,151,131),(3,107,38,59,88,69,152,130),(4,106,39,58,89,68,153,129),(5,105,40,57,90,67,154,128),(6,104,21,56,91,66,155,127),(7,103,22,55,92,65,156,126),(8,102,23,54,93,64,157,125),(9,101,24,53,94,63,158,124),(10,120,25,52,95,62,159,123),(11,119,26,51,96,61,160,122),(12,118,27,50,97,80,141,121),(13,117,28,49,98,79,142,140),(14,116,29,48,99,78,143,139),(15,115,30,47,100,77,144,138),(16,114,31,46,81,76,145,137),(17,113,32,45,82,75,146,136),(18,112,33,44,83,74,147,135),(19,111,34,43,84,73,148,134),(20,110,35,42,85,72,149,133)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,155),(22,144),(23,153),(24,142),(25,151),(26,160),(27,149),(28,158),(29,147),(30,156),(31,145),(32,154),(33,143),(34,152),(35,141),(36,150),(37,159),(38,148),(39,157),(40,146),(41,109),(42,118),(43,107),(44,116),(45,105),(46,114),(47,103),(48,112),(49,101),(50,110),(51,119),(52,108),(53,117),(54,106),(55,115),(56,104),(57,113),(58,102),(59,111),(60,120),(61,122),(62,131),(63,140),(64,129),(65,138),(66,127),(67,136),(68,125),(69,134),(70,123),(71,132),(72,121),(73,130),(74,139),(75,128),(76,137),(77,126),(78,135),(79,124),(80,133),(82,90),(83,99),(84,88),(85,97),(87,95),(89,93),(92,100),(94,98)]])
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 | 20A | ··· | 20L | 20M | ··· | 20T |
| 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 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 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 | D4 | D4 | D5 | SD16 | C4○D4 | D10 | D10 | C5⋊D4 | C8.C22 | Q8⋊D5 | D4×D5 | D4⋊2D5 | C20.C23 |
| kernel | C20⋊5SD16 | C20⋊3C8 | Q8⋊Dic5 | C4×D20 | C2×Q8⋊D5 | C5×C4⋊Q8 | D20 | C2×C20 | C4⋊Q8 | C20 | C20 | C42 | C2×Q8 | C2×C4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C20⋊5SD16 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 18 |
| 0 | 0 | 0 | 0 | 29 | 25 |
| 15 | 15 | 0 | 0 | 0 | 0 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 24 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 37 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,34,0,0,0,0,0,0,16,29,0,0,0,0,18,25],[15,26,0,0,0,0,15,15,0,0,0,0,0,0,3,24,0,0,0,0,3,38,0,0,0,0,0,0,29,5,0,0,0,0,37,12],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,34,40,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C20⋊5SD16 in GAP, Magma, Sage, TeX
C_{20}\rtimes_5{\rm SD}_{16} % in TeX
G:=Group("C20:5SD16"); // GroupNames label
G:=SmallGroup(320,710);
// by ID
G=gap.SmallGroup(320,710);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^3>;
// generators/relations