direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C5⋊SD32, Q16⋊6D10, C10⋊3SD32, C20.23D8, C40.24D4, C40.26C23, D40.9C22, C5⋊4(C2×SD32), (C2×Q16)⋊1D5, (C10×Q16)⋊4C2, C10.65(C2×D8), (C2×C10).44D8, C4.10(D4⋊D5), C5⋊2C16⋊9C22, (C2×D40).10C2, (C2×C8).239D10, (C2×C20).182D4, C20.181(C2×D4), C8.16(C5⋊D4), (C5×Q16)⋊6C22, C8.32(C22×D5), (C2×C40).91C22, C22.23(D4⋊D5), (C2×C5⋊2C16)⋊8C2, C2.20(C2×D4⋊D5), C4.11(C2×C5⋊D4), (C2×C4).144(C5⋊D4), SmallGroup(320,805)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C5⋊SD32
G = < a,b,c,d | a2=b5=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c7 >
Subgroups: 478 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C2×C8, D8, Q16, Q16, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C2×C16, SD32, C2×D8, C2×Q16, C40, D20, C2×C20, C2×C20, C5×Q8, C22×D5, C2×SD32, C5⋊2C16, D40, D40, C2×C40, C5×Q16, C5×Q16, C2×D20, Q8×C10, C2×C5⋊2C16, C5⋊SD32, C2×D40, C10×Q16, C2×C5⋊SD32
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, SD32, C2×D8, C5⋊D4, C22×D5, C2×SD32, D4⋊D5, C2×C5⋊D4, C5⋊SD32, C2×D4⋊D5, C2×C5⋊SD32
►On 160 points
Generators in S160
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 129)(13 130)(14 131)(15 132)(16 133)(17 78)(18 79)(19 80)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(81 126)(82 127)(83 128)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)
(1 58 155 91 24)(2 25 92 156 59)(3 60 157 93 26)(4 27 94 158 61)(5 62 159 95 28)(6 29 96 160 63)(7 64 145 81 30)(8 31 82 146 49)(9 50 147 83 32)(10 17 84 148 51)(11 52 149 85 18)(12 19 86 150 53)(13 54 151 87 20)(14 21 88 152 55)(15 56 153 89 22)(16 23 90 154 57)(33 135 70 121 100)(34 101 122 71 136)(35 137 72 123 102)(36 103 124 73 138)(37 139 74 125 104)(38 105 126 75 140)(39 141 76 127 106)(40 107 128 77 142)(41 143 78 113 108)(42 109 114 79 144)(43 129 80 115 110)(44 111 116 65 130)(45 131 66 117 112)(46 97 118 67 132)(47 133 68 119 98)(48 99 120 69 134)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 57)(18 64)(19 55)(20 62)(21 53)(22 60)(23 51)(24 58)(25 49)(26 56)(27 63)(28 54)(29 61)(30 52)(31 59)(32 50)(33 76)(34 67)(35 74)(36 65)(37 72)(38 79)(39 70)(40 77)(41 68)(42 75)(43 66)(44 73)(45 80)(46 71)(47 78)(48 69)(81 149)(82 156)(83 147)(84 154)(85 145)(86 152)(87 159)(88 150)(89 157)(90 148)(91 155)(92 146)(93 153)(94 160)(95 151)(96 158)(97 122)(98 113)(99 120)(100 127)(101 118)(102 125)(103 116)(104 123)(105 114)(106 121)(107 128)(108 119)(109 126)(110 117)(111 124)(112 115)(129 131)(130 138)(132 136)(133 143)(135 141)(137 139)(140 144)
G:=sub<Sym(160)| (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,129)(13,130)(14,131)(15,132)(16,133)(17,78)(18,79)(19,80)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(81,126)(82,127)(83,128)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,153)(98,154)(99,155)(100,156)(101,157)(102,158)(103,159)(104,160)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152), (1,58,155,91,24)(2,25,92,156,59)(3,60,157,93,26)(4,27,94,158,61)(5,62,159,95,28)(6,29,96,160,63)(7,64,145,81,30)(8,31,82,146,49)(9,50,147,83,32)(10,17,84,148,51)(11,52,149,85,18)(12,19,86,150,53)(13,54,151,87,20)(14,21,88,152,55)(15,56,153,89,22)(16,23,90,154,57)(33,135,70,121,100)(34,101,122,71,136)(35,137,72,123,102)(36,103,124,73,138)(37,139,74,125,104)(38,105,126,75,140)(39,141,76,127,106)(40,107,128,77,142)(41,143,78,113,108)(42,109,114,79,144)(43,129,80,115,110)(44,111,116,65,130)(45,131,66,117,112)(46,97,118,67,132)(47,133,68,119,98)(48,99,120,69,134), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,57)(18,64)(19,55)(20,62)(21,53)(22,60)(23,51)(24,58)(25,49)(26,56)(27,63)(28,54)(29,61)(30,52)(31,59)(32,50)(33,76)(34,67)(35,74)(36,65)(37,72)(38,79)(39,70)(40,77)(41,68)(42,75)(43,66)(44,73)(45,80)(46,71)(47,78)(48,69)(81,149)(82,156)(83,147)(84,154)(85,145)(86,152)(87,159)(88,150)(89,157)(90,148)(91,155)(92,146)(93,153)(94,160)(95,151)(96,158)(97,122)(98,113)(99,120)(100,127)(101,118)(102,125)(103,116)(104,123)(105,114)(106,121)(107,128)(108,119)(109,126)(110,117)(111,124)(112,115)(129,131)(130,138)(132,136)(133,143)(135,141)(137,139)(140,144)>;
G:=Group( (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,129)(13,130)(14,131)(15,132)(16,133)(17,78)(18,79)(19,80)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(81,126)(82,127)(83,128)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,153)(98,154)(99,155)(100,156)(101,157)(102,158)(103,159)(104,160)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152), (1,58,155,91,24)(2,25,92,156,59)(3,60,157,93,26)(4,27,94,158,61)(5,62,159,95,28)(6,29,96,160,63)(7,64,145,81,30)(8,31,82,146,49)(9,50,147,83,32)(10,17,84,148,51)(11,52,149,85,18)(12,19,86,150,53)(13,54,151,87,20)(14,21,88,152,55)(15,56,153,89,22)(16,23,90,154,57)(33,135,70,121,100)(34,101,122,71,136)(35,137,72,123,102)(36,103,124,73,138)(37,139,74,125,104)(38,105,126,75,140)(39,141,76,127,106)(40,107,128,77,142)(41,143,78,113,108)(42,109,114,79,144)(43,129,80,115,110)(44,111,116,65,130)(45,131,66,117,112)(46,97,118,67,132)(47,133,68,119,98)(48,99,120,69,134), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,57)(18,64)(19,55)(20,62)(21,53)(22,60)(23,51)(24,58)(25,49)(26,56)(27,63)(28,54)(29,61)(30,52)(31,59)(32,50)(33,76)(34,67)(35,74)(36,65)(37,72)(38,79)(39,70)(40,77)(41,68)(42,75)(43,66)(44,73)(45,80)(46,71)(47,78)(48,69)(81,149)(82,156)(83,147)(84,154)(85,145)(86,152)(87,159)(88,150)(89,157)(90,148)(91,155)(92,146)(93,153)(94,160)(95,151)(96,158)(97,122)(98,113)(99,120)(100,127)(101,118)(102,125)(103,116)(104,123)(105,114)(106,121)(107,128)(108,119)(109,126)(110,117)(111,124)(112,115)(129,131)(130,138)(132,136)(133,143)(135,141)(137,139)(140,144) );
G=PermutationGroup([[(1,134),(2,135),(3,136),(4,137),(5,138),(6,139),(7,140),(8,141),(9,142),(10,143),(11,144),(12,129),(13,130),(14,131),(15,132),(16,133),(17,78),(18,79),(19,80),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(81,126),(82,127),(83,128),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119),(91,120),(92,121),(93,122),(94,123),(95,124),(96,125),(97,153),(98,154),(99,155),(100,156),(101,157),(102,158),(103,159),(104,160),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152)], [(1,58,155,91,24),(2,25,92,156,59),(3,60,157,93,26),(4,27,94,158,61),(5,62,159,95,28),(6,29,96,160,63),(7,64,145,81,30),(8,31,82,146,49),(9,50,147,83,32),(10,17,84,148,51),(11,52,149,85,18),(12,19,86,150,53),(13,54,151,87,20),(14,21,88,152,55),(15,56,153,89,22),(16,23,90,154,57),(33,135,70,121,100),(34,101,122,71,136),(35,137,72,123,102),(36,103,124,73,138),(37,139,74,125,104),(38,105,126,75,140),(39,141,76,127,106),(40,107,128,77,142),(41,143,78,113,108),(42,109,114,79,144),(43,129,80,115,110),(44,111,116,65,130),(45,131,66,117,112),(46,97,118,67,132),(47,133,68,119,98),(48,99,120,69,134)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,57),(18,64),(19,55),(20,62),(21,53),(22,60),(23,51),(24,58),(25,49),(26,56),(27,63),(28,54),(29,61),(30,52),(31,59),(32,50),(33,76),(34,67),(35,74),(36,65),(37,72),(38,79),(39,70),(40,77),(41,68),(42,75),(43,66),(44,73),(45,80),(46,71),(47,78),(48,69),(81,149),(82,156),(83,147),(84,154),(85,145),(86,152),(87,159),(88,150),(89,157),(90,148),(91,155),(92,146),(93,153),(94,160),(95,151),(96,158),(97,122),(98,113),(99,120),(100,127),(101,118),(102,125),(103,116),(104,123),(105,114),(106,121),(107,128),(108,119),(109,126),(110,117),(111,124),(112,115),(129,131),(130,138),(132,136),(133,143),(135,141),(137,139),(140,144)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | SD32 | C5⋊D4 | C5⋊D4 | D4⋊D5 | D4⋊D5 | C5⋊SD32 |
| kernel | C2×C5⋊SD32 | C2×C5⋊2C16 | C5⋊SD32 | C2×D40 | C10×Q16 | C40 | C2×C20 | C2×Q16 | C20 | C2×C10 | C2×C8 | Q16 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 2 | 2 | 8 |
Matrix representation of C2×C5⋊SD32 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 189 | 240 |
| 0 | 0 | 1 | 0 |
| 97 | 160 | 0 | 0 |
| 223 | 179 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 52 | 1 |
| 1 | 0 | 0 | 0 |
| 118 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 189 | 240 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,189,1,0,0,240,0],[97,223,0,0,160,179,0,0,0,0,240,52,0,0,0,1],[1,118,0,0,0,240,0,0,0,0,1,189,0,0,0,240] >;
C2×C5⋊SD32 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes {\rm SD}_{32} % in TeX
G:=Group("C2xC5:SD32"); // GroupNames label
G:=SmallGroup(320,805);
// by ID
G=gap.SmallGroup(320,805);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,184,675,185,192,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^7>;
// generators/relations