metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊4SD16, C42.219D10, C5⋊2C8⋊12D4, C5⋊2(C8⋊5D4), C4.16(D4×D5), C4⋊1(D4.D5), C4⋊1D4.6D5, C20.34(C2×D4), C20⋊2Q8⋊20C2, (C2×D4).59D10, (C2×C20).150D4, C10.59(C2×SD16), C2.13(C20⋊D4), C10.22(C4⋊1D4), (C2×C20).394C23, (C4×C20).124C22, (D4×C10).75C22, (C2×Dic10).116C22, (C4×C5⋊2C8)⋊16C2, (C2×D4.D5)⋊15C2, (C5×C4⋊1D4).5C2, C2.13(C2×D4.D5), (C2×C10).525(C2×D4), (C2×C4).132(C5⋊D4), (C2×C4).492(C22×D5), C22.198(C2×C5⋊D4), (C2×C5⋊2C8).266C22, SmallGroup(320,703)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊4SD16
G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a11, cbc=b3 >
Subgroups: 510 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C8⋊5D4, C2×C5⋊2C8, C4⋊Dic5, D4.D5, C4×C20, C2×Dic10, D4×C10, D4×C10, C4×C5⋊2C8, C20⋊2Q8, C2×D4.D5, C5×C4⋊1D4, C20⋊4SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C4⋊1D4, C2×SD16, C5⋊D4, C22×D5, C8⋊5D4, D4.D5, D4×D5, C2×C5⋊D4, C2×D4.D5, C20⋊D4, C20⋊4SD16
►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)
(1 120 28 81 70 127 154 48)(2 109 29 90 71 136 155 57)(3 118 30 99 72 125 156 46)(4 107 31 88 73 134 157 55)(5 116 32 97 74 123 158 44)(6 105 33 86 75 132 159 53)(7 114 34 95 76 121 160 42)(8 103 35 84 77 130 141 51)(9 112 36 93 78 139 142 60)(10 101 37 82 79 128 143 49)(11 110 38 91 80 137 144 58)(12 119 39 100 61 126 145 47)(13 108 40 89 62 135 146 56)(14 117 21 98 63 124 147 45)(15 106 22 87 64 133 148 54)(16 115 23 96 65 122 149 43)(17 104 24 85 66 131 150 52)(18 113 25 94 67 140 151 41)(19 102 26 83 68 129 152 50)(20 111 27 92 69 138 153 59)
(1 75)(2 66)(3 77)(4 68)(5 79)(6 70)(7 61)(8 72)(9 63)(10 74)(11 65)(12 76)(13 67)(14 78)(15 69)(16 80)(17 71)(18 62)(19 73)(20 64)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)(41 108)(42 119)(43 110)(44 101)(45 112)(46 103)(47 114)(48 105)(49 116)(50 107)(51 118)(52 109)(53 120)(54 111)(55 102)(56 113)(57 104)(58 115)(59 106)(60 117)(81 132)(82 123)(83 134)(84 125)(85 136)(86 127)(87 138)(88 129)(89 140)(90 131)(91 122)(92 133)(93 124)(94 135)(95 126)(96 137)(97 128)(98 139)(99 130)(100 121)(141 156)(142 147)(143 158)(144 149)(145 160)(146 151)(148 153)(150 155)(152 157)(154 159)
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), (1,120,28,81,70,127,154,48)(2,109,29,90,71,136,155,57)(3,118,30,99,72,125,156,46)(4,107,31,88,73,134,157,55)(5,116,32,97,74,123,158,44)(6,105,33,86,75,132,159,53)(7,114,34,95,76,121,160,42)(8,103,35,84,77,130,141,51)(9,112,36,93,78,139,142,60)(10,101,37,82,79,128,143,49)(11,110,38,91,80,137,144,58)(12,119,39,100,61,126,145,47)(13,108,40,89,62,135,146,56)(14,117,21,98,63,124,147,45)(15,106,22,87,64,133,148,54)(16,115,23,96,65,122,149,43)(17,104,24,85,66,131,150,52)(18,113,25,94,67,140,151,41)(19,102,26,83,68,129,152,50)(20,111,27,92,69,138,153,59), (1,75)(2,66)(3,77)(4,68)(5,79)(6,70)(7,61)(8,72)(9,63)(10,74)(11,65)(12,76)(13,67)(14,78)(15,69)(16,80)(17,71)(18,62)(19,73)(20,64)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,108)(42,119)(43,110)(44,101)(45,112)(46,103)(47,114)(48,105)(49,116)(50,107)(51,118)(52,109)(53,120)(54,111)(55,102)(56,113)(57,104)(58,115)(59,106)(60,117)(81,132)(82,123)(83,134)(84,125)(85,136)(86,127)(87,138)(88,129)(89,140)(90,131)(91,122)(92,133)(93,124)(94,135)(95,126)(96,137)(97,128)(98,139)(99,130)(100,121)(141,156)(142,147)(143,158)(144,149)(145,160)(146,151)(148,153)(150,155)(152,157)(154,159)>;
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), (1,120,28,81,70,127,154,48)(2,109,29,90,71,136,155,57)(3,118,30,99,72,125,156,46)(4,107,31,88,73,134,157,55)(5,116,32,97,74,123,158,44)(6,105,33,86,75,132,159,53)(7,114,34,95,76,121,160,42)(8,103,35,84,77,130,141,51)(9,112,36,93,78,139,142,60)(10,101,37,82,79,128,143,49)(11,110,38,91,80,137,144,58)(12,119,39,100,61,126,145,47)(13,108,40,89,62,135,146,56)(14,117,21,98,63,124,147,45)(15,106,22,87,64,133,148,54)(16,115,23,96,65,122,149,43)(17,104,24,85,66,131,150,52)(18,113,25,94,67,140,151,41)(19,102,26,83,68,129,152,50)(20,111,27,92,69,138,153,59), (1,75)(2,66)(3,77)(4,68)(5,79)(6,70)(7,61)(8,72)(9,63)(10,74)(11,65)(12,76)(13,67)(14,78)(15,69)(16,80)(17,71)(18,62)(19,73)(20,64)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,108)(42,119)(43,110)(44,101)(45,112)(46,103)(47,114)(48,105)(49,116)(50,107)(51,118)(52,109)(53,120)(54,111)(55,102)(56,113)(57,104)(58,115)(59,106)(60,117)(81,132)(82,123)(83,134)(84,125)(85,136)(86,127)(87,138)(88,129)(89,140)(90,131)(91,122)(92,133)(93,124)(94,135)(95,126)(96,137)(97,128)(98,139)(99,130)(100,121)(141,156)(142,147)(143,158)(144,149)(145,160)(146,151)(148,153)(150,155)(152,157)(154,159) );
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)], [(1,120,28,81,70,127,154,48),(2,109,29,90,71,136,155,57),(3,118,30,99,72,125,156,46),(4,107,31,88,73,134,157,55),(5,116,32,97,74,123,158,44),(6,105,33,86,75,132,159,53),(7,114,34,95,76,121,160,42),(8,103,35,84,77,130,141,51),(9,112,36,93,78,139,142,60),(10,101,37,82,79,128,143,49),(11,110,38,91,80,137,144,58),(12,119,39,100,61,126,145,47),(13,108,40,89,62,135,146,56),(14,117,21,98,63,124,147,45),(15,106,22,87,64,133,148,54),(16,115,23,96,65,122,149,43),(17,104,24,85,66,131,150,52),(18,113,25,94,67,140,151,41),(19,102,26,83,68,129,152,50),(20,111,27,92,69,138,153,59)], [(1,75),(2,66),(3,77),(4,68),(5,79),(6,70),(7,61),(8,72),(9,63),(10,74),(11,65),(12,76),(13,67),(14,78),(15,69),(16,80),(17,71),(18,62),(19,73),(20,64),(21,36),(22,27),(23,38),(24,29),(25,40),(26,31),(28,33),(30,35),(32,37),(34,39),(41,108),(42,119),(43,110),(44,101),(45,112),(46,103),(47,114),(48,105),(49,116),(50,107),(51,118),(52,109),(53,120),(54,111),(55,102),(56,113),(57,104),(58,115),(59,106),(60,117),(81,132),(82,123),(83,134),(84,125),(85,136),(86,127),(87,138),(88,129),(89,140),(90,131),(91,122),(92,133),(93,124),(94,135),(95,126),(96,137),(97,128),(98,139),(99,130),(100,121),(141,156),(142,147),(143,158),(144,149),(145,160),(146,151),(148,153),(150,155),(152,157),(154,159)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 40 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | D10 | D10 | C5⋊D4 | D4.D5 | D4×D5 |
| kernel | C20⋊4SD16 | C4×C5⋊2C8 | C20⋊2Q8 | C2×D4.D5 | C5×C4⋊1D4 | C5⋊2C8 | C2×C20 | C4⋊1D4 | C20 | C42 | C2×D4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 2 | 8 | 2 | 4 | 8 | 8 | 4 |
Matrix representation of C20⋊4SD16 ►in GL6(𝔽41)
| 37 | 30 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 40 | 16 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 25 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [37,0,0,0,0,0,30,10,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[40,5,0,0,0,0,16,1,0,0,0,0,0,0,0,9,0,0,0,0,32,30,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,25,40,0,0,0,0,0,0,1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,39,40] >;
C20⋊4SD16 in GAP, Magma, Sage, TeX
C_{20}\rtimes_4{\rm SD}_{16} % in TeX
G:=Group("C20:4SD16"); // GroupNames label
G:=SmallGroup(320,703);
// by ID
G=gap.SmallGroup(320,703);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^11,c*b*c=b^3>;
// generators/relations