metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8⋊23D4, C5⋊4(C8⋊8D4), C4⋊C4.61D10, (C2×C10)⋊4SD16, C4⋊D4.6D5, C4.172(D4×D5), (C2×D4).41D10, C20.150(C2×D4), (C2×C20).264D4, C10.98(C4○D8), C10.Q16⋊35C2, D4⋊Dic5⋊17C2, C22⋊1(D4.D5), C20.Q8⋊36C2, C10.56(C2×SD16), (C22×C10).87D4, C20.185(C4○D4), C20.48D4⋊24C2, C4.61(D4⋊2D5), C10.95(C4⋊D4), (C2×C20).360C23, (D4×C10).57C22, (C22×C4).341D10, C23.40(C5⋊D4), C4⋊Dic5.144C22, C2.16(Dic5⋊D4), C2.17(D4.8D10), (C22×C20).164C22, (C2×Dic10).107C22, (C2×D4.D5)⋊10C2, (C22×C5⋊2C8)⋊4C2, (C5×C4⋊D4).5C2, C2.10(C2×D4.D5), (C2×C10).491(C2×D4), (C2×C4).106(C5⋊D4), (C5×C4⋊C4).108C22, (C2×C4).460(C22×D5), C22.166(C2×C5⋊D4), (C2×C5⋊2C8).257C22, SmallGroup(320,668)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊2C8⋊23D4
G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b3, bd=db, dcd=c-1 >
Subgroups: 406 in 124 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C5⋊2C8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C8⋊8D4, C2×C5⋊2C8, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D4.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, D4×C10, C20.Q8, C10.Q16, D4⋊Dic5, C22×C5⋊2C8, C20.48D4, C2×D4.D5, C5×C4⋊D4, C5⋊2C8⋊23D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, C5⋊D4, C22×D5, C8⋊8D4, D4.D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C2×D4.D5, Dic5⋊D4, D4.8D10, C5⋊2C8⋊23D4
►On 160 points
Generators in S160
(1 74 38 47 155)(2 156 48 39 75)(3 76 40 41 157)(4 158 42 33 77)(5 78 34 43 159)(6 160 44 35 79)(7 80 36 45 153)(8 154 46 37 73)(9 85 103 144 62)(10 63 137 104 86)(11 87 97 138 64)(12 57 139 98 88)(13 81 99 140 58)(14 59 141 100 82)(15 83 101 142 60)(16 61 143 102 84)(17 93 111 131 70)(18 71 132 112 94)(19 95 105 133 72)(20 65 134 106 96)(21 89 107 135 66)(22 67 136 108 90)(23 91 109 129 68)(24 69 130 110 92)(25 54 151 121 113)(26 114 122 152 55)(27 56 145 123 115)(28 116 124 146 49)(29 50 147 125 117)(30 118 126 148 51)(31 52 149 127 119)(32 120 128 150 53)
(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 96 88 151)(2 91 81 146)(3 94 82 149)(4 89 83 152)(5 92 84 147)(6 95 85 150)(7 90 86 145)(8 93 87 148)(9 128 79 19)(10 123 80 22)(11 126 73 17)(12 121 74 20)(13 124 75 23)(14 127 76 18)(15 122 77 21)(16 125 78 24)(25 47 134 139)(26 42 135 142)(27 45 136 137)(28 48 129 140)(29 43 130 143)(30 46 131 138)(31 41 132 141)(32 44 133 144)(33 66 60 114)(34 69 61 117)(35 72 62 120)(36 67 63 115)(37 70 64 118)(38 65 57 113)(39 68 58 116)(40 71 59 119)(49 156 109 99)(50 159 110 102)(51 154 111 97)(52 157 112 100)(53 160 105 103)(54 155 106 98)(55 158 107 101)(56 153 108 104)
(1 151)(2 152)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 116)(34 117)(35 118)(36 119)(37 120)(38 113)(39 114)(40 115)(49 158)(50 159)(51 160)(52 153)(53 154)(54 155)(55 156)(56 157)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 71)(64 72)(73 128)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)(129 142)(130 143)(131 144)(132 137)(133 138)(134 139)(135 140)(136 141)
G:=sub<Sym(160)| (1,74,38,47,155)(2,156,48,39,75)(3,76,40,41,157)(4,158,42,33,77)(5,78,34,43,159)(6,160,44,35,79)(7,80,36,45,153)(8,154,46,37,73)(9,85,103,144,62)(10,63,137,104,86)(11,87,97,138,64)(12,57,139,98,88)(13,81,99,140,58)(14,59,141,100,82)(15,83,101,142,60)(16,61,143,102,84)(17,93,111,131,70)(18,71,132,112,94)(19,95,105,133,72)(20,65,134,106,96)(21,89,107,135,66)(22,67,136,108,90)(23,91,109,129,68)(24,69,130,110,92)(25,54,151,121,113)(26,114,122,152,55)(27,56,145,123,115)(28,116,124,146,49)(29,50,147,125,117)(30,118,126,148,51)(31,52,149,127,119)(32,120,128,150,53), (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), (1,96,88,151)(2,91,81,146)(3,94,82,149)(4,89,83,152)(5,92,84,147)(6,95,85,150)(7,90,86,145)(8,93,87,148)(9,128,79,19)(10,123,80,22)(11,126,73,17)(12,121,74,20)(13,124,75,23)(14,127,76,18)(15,122,77,21)(16,125,78,24)(25,47,134,139)(26,42,135,142)(27,45,136,137)(28,48,129,140)(29,43,130,143)(30,46,131,138)(31,41,132,141)(32,44,133,144)(33,66,60,114)(34,69,61,117)(35,72,62,120)(36,67,63,115)(37,70,64,118)(38,65,57,113)(39,68,58,116)(40,71,59,119)(49,156,109,99)(50,159,110,102)(51,154,111,97)(52,157,112,100)(53,160,105,103)(54,155,106,98)(55,158,107,101)(56,153,108,104), (1,151)(2,152)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,116)(34,117)(35,118)(36,119)(37,120)(38,113)(39,114)(40,115)(49,158)(50,159)(51,160)(52,153)(53,154)(54,155)(55,156)(56,157)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(73,128)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(129,142)(130,143)(131,144)(132,137)(133,138)(134,139)(135,140)(136,141)>;
G:=Group( (1,74,38,47,155)(2,156,48,39,75)(3,76,40,41,157)(4,158,42,33,77)(5,78,34,43,159)(6,160,44,35,79)(7,80,36,45,153)(8,154,46,37,73)(9,85,103,144,62)(10,63,137,104,86)(11,87,97,138,64)(12,57,139,98,88)(13,81,99,140,58)(14,59,141,100,82)(15,83,101,142,60)(16,61,143,102,84)(17,93,111,131,70)(18,71,132,112,94)(19,95,105,133,72)(20,65,134,106,96)(21,89,107,135,66)(22,67,136,108,90)(23,91,109,129,68)(24,69,130,110,92)(25,54,151,121,113)(26,114,122,152,55)(27,56,145,123,115)(28,116,124,146,49)(29,50,147,125,117)(30,118,126,148,51)(31,52,149,127,119)(32,120,128,150,53), (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), (1,96,88,151)(2,91,81,146)(3,94,82,149)(4,89,83,152)(5,92,84,147)(6,95,85,150)(7,90,86,145)(8,93,87,148)(9,128,79,19)(10,123,80,22)(11,126,73,17)(12,121,74,20)(13,124,75,23)(14,127,76,18)(15,122,77,21)(16,125,78,24)(25,47,134,139)(26,42,135,142)(27,45,136,137)(28,48,129,140)(29,43,130,143)(30,46,131,138)(31,41,132,141)(32,44,133,144)(33,66,60,114)(34,69,61,117)(35,72,62,120)(36,67,63,115)(37,70,64,118)(38,65,57,113)(39,68,58,116)(40,71,59,119)(49,156,109,99)(50,159,110,102)(51,154,111,97)(52,157,112,100)(53,160,105,103)(54,155,106,98)(55,158,107,101)(56,153,108,104), (1,151)(2,152)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,116)(34,117)(35,118)(36,119)(37,120)(38,113)(39,114)(40,115)(49,158)(50,159)(51,160)(52,153)(53,154)(54,155)(55,156)(56,157)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(73,128)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(129,142)(130,143)(131,144)(132,137)(133,138)(134,139)(135,140)(136,141) );
G=PermutationGroup([[(1,74,38,47,155),(2,156,48,39,75),(3,76,40,41,157),(4,158,42,33,77),(5,78,34,43,159),(6,160,44,35,79),(7,80,36,45,153),(8,154,46,37,73),(9,85,103,144,62),(10,63,137,104,86),(11,87,97,138,64),(12,57,139,98,88),(13,81,99,140,58),(14,59,141,100,82),(15,83,101,142,60),(16,61,143,102,84),(17,93,111,131,70),(18,71,132,112,94),(19,95,105,133,72),(20,65,134,106,96),(21,89,107,135,66),(22,67,136,108,90),(23,91,109,129,68),(24,69,130,110,92),(25,54,151,121,113),(26,114,122,152,55),(27,56,145,123,115),(28,116,124,146,49),(29,50,147,125,117),(30,118,126,148,51),(31,52,149,127,119),(32,120,128,150,53)], [(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)], [(1,96,88,151),(2,91,81,146),(3,94,82,149),(4,89,83,152),(5,92,84,147),(6,95,85,150),(7,90,86,145),(8,93,87,148),(9,128,79,19),(10,123,80,22),(11,126,73,17),(12,121,74,20),(13,124,75,23),(14,127,76,18),(15,122,77,21),(16,125,78,24),(25,47,134,139),(26,42,135,142),(27,45,136,137),(28,48,129,140),(29,43,130,143),(30,46,131,138),(31,41,132,141),(32,44,133,144),(33,66,60,114),(34,69,61,117),(35,72,62,120),(36,67,63,115),(37,70,64,118),(38,65,57,113),(39,68,58,116),(40,71,59,119),(49,156,109,99),(50,159,110,102),(51,154,111,97),(52,157,112,100),(53,160,105,103),(54,155,106,98),(55,158,107,101),(56,153,108,104)], [(1,151),(2,152),(3,145),(4,146),(5,147),(6,148),(7,149),(8,150),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,116),(34,117),(35,118),(36,119),(37,120),(38,113),(39,114),(40,115),(49,158),(50,159),(51,160),(52,153),(53,154),(54,155),(55,156),(56,157),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,71),(64,72),(73,128),(74,121),(75,122),(76,123),(77,124),(78,125),(79,126),(80,127),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(97,105),(98,106),(99,107),(100,108),(101,109),(102,110),(103,111),(104,112),(129,142),(130,143),(131,144),(132,137),(133,138),(134,139),(135,140),(136,141)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 8 | 40 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | SD16 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | D4×D5 | D4⋊2D5 | D4.D5 | D4.8D10 |
| kernel | C5⋊2C8⋊23D4 | C20.Q8 | C10.Q16 | D4⋊Dic5 | C22×C5⋊2C8 | C20.48D4 | C2×D4.D5 | C5×C4⋊D4 | C5⋊2C8 | C2×C20 | C22×C10 | C4⋊D4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C2×C4 | C23 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of C5⋊2C8⋊23D4 ►in GL6(𝔽41)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 14 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 31 | 0 | 0 | 0 | 0 |
| 35 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 15 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 23 |
| 0 | 0 | 0 | 0 | 25 | 11 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 14 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 18 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 39 |
| 0 | 0 | 0 | 0 | 22 | 24 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 18 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 39 |
| 0 | 0 | 0 | 0 | 21 | 24 |
G:=sub<GL(6,GF(41))| [16,14,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,35,0,0,0,0,31,29,0,0,0,0,0,0,0,15,0,0,0,0,11,11,0,0,0,0,0,0,30,25,0,0,0,0,23,11],[40,14,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,18,9,0,0,0,0,0,0,17,22,0,0,0,0,39,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,32,0,0,0,0,18,9,0,0,0,0,0,0,17,21,0,0,0,0,39,24] >;
C5⋊2C8⋊23D4 in GAP, Magma, Sage, TeX
C_5\rtimes_2C_8\rtimes_{23}D_4 % in TeX
G:=Group("C5:2C8:23D4"); // GroupNames label
G:=SmallGroup(320,668);
// by ID
G=gap.SmallGroup(320,668);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,d*c*d=c^-1>;
// generators/relations