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