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 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, C10 [×3], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×5], D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C5⋊2C8 [×2], C5⋊2C8, Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×C10, C22×C10, C8⋊8D4, C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×2], C10.D4, C4⋊Dic5, D4.D5 [×2], 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 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, C5⋊D4 [×2], C22×D5, C8⋊8D4, D4.D5 [×2], 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 37 47 155)(2 156 48 38 75)(3 76 39 41 157)(4 158 42 40 77)(5 78 33 43 159)(6 160 44 34 79)(7 80 35 45 153)(8 154 46 36 73)(9 126 103 131 142)(10 143 132 104 127)(11 128 97 133 144)(12 137 134 98 121)(13 122 99 135 138)(14 139 136 100 123)(15 124 101 129 140)(16 141 130 102 125)(17 62 49 68 94)(18 95 69 50 63)(19 64 51 70 96)(20 89 71 52 57)(21 58 53 72 90)(22 91 65 54 59)(23 60 55 66 92)(24 93 67 56 61)(25 84 149 106 113)(26 114 107 150 85)(27 86 151 108 115)(28 116 109 152 87)(29 88 145 110 117)(30 118 111 146 81)(31 82 147 112 119)(32 120 105 148 83)
(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)
(1 113 121 50)(2 116 122 53)(3 119 123 56)(4 114 124 51)(5 117 125 54)(6 120 126 49)(7 115 127 52)(8 118 128 55)(9 62 79 32)(10 57 80 27)(11 60 73 30)(12 63 74 25)(13 58 75 28)(14 61 76 31)(15 64 77 26)(16 59 78 29)(17 34 83 142)(18 37 84 137)(19 40 85 140)(20 35 86 143)(21 38 87 138)(22 33 88 141)(23 36 81 144)(24 39 82 139)(41 147 136 93)(42 150 129 96)(43 145 130 91)(44 148 131 94)(45 151 132 89)(46 146 133 92)(47 149 134 95)(48 152 135 90)(65 159 110 102)(66 154 111 97)(67 157 112 100)(68 160 105 103)(69 155 106 98)(70 158 107 101)(71 153 108 104)(72 156 109 99)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(57 76)(58 77)(59 78)(60 79)(61 80)(62 73)(63 74)(64 75)(65 159)(66 160)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(81 142)(82 143)(83 144)(84 137)(85 138)(86 139)(87 140)(88 141)(97 105)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(129 152)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)
G:=sub<Sym(160)| (1,74,37,47,155)(2,156,48,38,75)(3,76,39,41,157)(4,158,42,40,77)(5,78,33,43,159)(6,160,44,34,79)(7,80,35,45,153)(8,154,46,36,73)(9,126,103,131,142)(10,143,132,104,127)(11,128,97,133,144)(12,137,134,98,121)(13,122,99,135,138)(14,139,136,100,123)(15,124,101,129,140)(16,141,130,102,125)(17,62,49,68,94)(18,95,69,50,63)(19,64,51,70,96)(20,89,71,52,57)(21,58,53,72,90)(22,91,65,54,59)(23,60,55,66,92)(24,93,67,56,61)(25,84,149,106,113)(26,114,107,150,85)(27,86,151,108,115)(28,116,109,152,87)(29,88,145,110,117)(30,118,111,146,81)(31,82,147,112,119)(32,120,105,148,83), (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), (1,113,121,50)(2,116,122,53)(3,119,123,56)(4,114,124,51)(5,117,125,54)(6,120,126,49)(7,115,127,52)(8,118,128,55)(9,62,79,32)(10,57,80,27)(11,60,73,30)(12,63,74,25)(13,58,75,28)(14,61,76,31)(15,64,77,26)(16,59,78,29)(17,34,83,142)(18,37,84,137)(19,40,85,140)(20,35,86,143)(21,38,87,138)(22,33,88,141)(23,36,81,144)(24,39,82,139)(41,147,136,93)(42,150,129,96)(43,145,130,91)(44,148,131,94)(45,151,132,89)(46,146,133,92)(47,149,134,95)(48,152,135,90)(65,159,110,102)(66,154,111,97)(67,157,112,100)(68,160,105,103)(69,155,106,98)(70,158,107,101)(71,153,108,104)(72,156,109,99), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(57,76)(58,77)(59,78)(60,79)(61,80)(62,73)(63,74)(64,75)(65,159)(66,160)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(81,142)(82,143)(83,144)(84,137)(85,138)(86,139)(87,140)(88,141)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(129,152)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151)>;
G:=Group( (1,74,37,47,155)(2,156,48,38,75)(3,76,39,41,157)(4,158,42,40,77)(5,78,33,43,159)(6,160,44,34,79)(7,80,35,45,153)(8,154,46,36,73)(9,126,103,131,142)(10,143,132,104,127)(11,128,97,133,144)(12,137,134,98,121)(13,122,99,135,138)(14,139,136,100,123)(15,124,101,129,140)(16,141,130,102,125)(17,62,49,68,94)(18,95,69,50,63)(19,64,51,70,96)(20,89,71,52,57)(21,58,53,72,90)(22,91,65,54,59)(23,60,55,66,92)(24,93,67,56,61)(25,84,149,106,113)(26,114,107,150,85)(27,86,151,108,115)(28,116,109,152,87)(29,88,145,110,117)(30,118,111,146,81)(31,82,147,112,119)(32,120,105,148,83), (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), (1,113,121,50)(2,116,122,53)(3,119,123,56)(4,114,124,51)(5,117,125,54)(6,120,126,49)(7,115,127,52)(8,118,128,55)(9,62,79,32)(10,57,80,27)(11,60,73,30)(12,63,74,25)(13,58,75,28)(14,61,76,31)(15,64,77,26)(16,59,78,29)(17,34,83,142)(18,37,84,137)(19,40,85,140)(20,35,86,143)(21,38,87,138)(22,33,88,141)(23,36,81,144)(24,39,82,139)(41,147,136,93)(42,150,129,96)(43,145,130,91)(44,148,131,94)(45,151,132,89)(46,146,133,92)(47,149,134,95)(48,152,135,90)(65,159,110,102)(66,154,111,97)(67,157,112,100)(68,160,105,103)(69,155,106,98)(70,158,107,101)(71,153,108,104)(72,156,109,99), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(57,76)(58,77)(59,78)(60,79)(61,80)(62,73)(63,74)(64,75)(65,159)(66,160)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(81,142)(82,143)(83,144)(84,137)(85,138)(86,139)(87,140)(88,141)(97,105)(98,106)(99,107)(100,108)(101,109)(102,110)(103,111)(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(129,152)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)(136,151) );
G=PermutationGroup([(1,74,37,47,155),(2,156,48,38,75),(3,76,39,41,157),(4,158,42,40,77),(5,78,33,43,159),(6,160,44,34,79),(7,80,35,45,153),(8,154,46,36,73),(9,126,103,131,142),(10,143,132,104,127),(11,128,97,133,144),(12,137,134,98,121),(13,122,99,135,138),(14,139,136,100,123),(15,124,101,129,140),(16,141,130,102,125),(17,62,49,68,94),(18,95,69,50,63),(19,64,51,70,96),(20,89,71,52,57),(21,58,53,72,90),(22,91,65,54,59),(23,60,55,66,92),(24,93,67,56,61),(25,84,149,106,113),(26,114,107,150,85),(27,86,151,108,115),(28,116,109,152,87),(29,88,145,110,117),(30,118,111,146,81),(31,82,147,112,119),(32,120,105,148,83)], [(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)], [(1,113,121,50),(2,116,122,53),(3,119,123,56),(4,114,124,51),(5,117,125,54),(6,120,126,49),(7,115,127,52),(8,118,128,55),(9,62,79,32),(10,57,80,27),(11,60,73,30),(12,63,74,25),(13,58,75,28),(14,61,76,31),(15,64,77,26),(16,59,78,29),(17,34,83,142),(18,37,84,137),(19,40,85,140),(20,35,86,143),(21,38,87,138),(22,33,88,141),(23,36,81,144),(24,39,82,139),(41,147,136,93),(42,150,129,96),(43,145,130,91),(44,148,131,94),(45,151,132,89),(46,146,133,92),(47,149,134,95),(48,152,135,90),(65,159,110,102),(66,154,111,97),(67,157,112,100),(68,160,105,103),(69,155,106,98),(70,158,107,101),(71,153,108,104),(72,156,109,99)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,49),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(57,76),(58,77),(59,78),(60,79),(61,80),(62,73),(63,74),(64,75),(65,159),(66,160),(67,153),(68,154),(69,155),(70,156),(71,157),(72,158),(81,142),(82,143),(83,144),(84,137),(85,138),(86,139),(87,140),(88,141),(97,105),(98,106),(99,107),(100,108),(101,109),(102,110),(103,111),(104,112),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128),(129,152),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150),(136,151)])
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