metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2C8.6D4, C4⋊C4.71D10, C4.177(D4×D5), (C2×C20).80D4, C5⋊3(C8.D4), C22⋊Q8.7D5, C20.158(C2×D4), (C2×Q8).33D10, Q8⋊Dic5⋊18C2, C10.Q16⋊40C2, C20.Q8⋊40C2, (C22×C10).98D4, C20.193(C4○D4), C4.66(D4⋊2D5), (C2×C20).371C23, (C22×C4).131D10, C23.29(C5⋊D4), (Q8×C10).51C22, C10.100(C4⋊D4), C20.48D4.13C2, C10.92(C8.C22), C4⋊Dic5.149C22, C2.21(Dic5⋊D4), C2.16(D4.9D10), C2.13(C20.C23), (C22×C20).175C22, (C2×Dic10).110C22, (C2×C5⋊Q16)⋊11C2, (C5×C22⋊Q8).6C2, (C2×C10).502(C2×D4), (C2×C4).58(C5⋊D4), (C5×C4⋊C4).118C22, (C2×C4).471(C22×D5), C22.177(C2×C5⋊D4), (C2×C4.Dic5).20C2, (C2×C5⋊2C8).118C22, SmallGroup(320,679)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊(C8.D4)
G = < a,b,c,d | a5=b8=c4=1, d2=b4, bab-1=a-1, ac=ca, ad=da, cbc-1=b3, dbd-1=b-1, dcd-1=b4c-1 >
Subgroups: 358 in 110 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4.Q8, C22⋊Q8, C22⋊Q8, C2×M4(2), C2×Q16, C5⋊2C8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×C10, C8.D4, C2×C5⋊2C8, C4.Dic5, C10.D4, C4⋊Dic5, C5⋊Q16, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, Q8×C10, C20.Q8, C10.Q16, Q8⋊Dic5, C2×C4.Dic5, C20.48D4, C2×C5⋊Q16, C5×C22⋊Q8, C5⋊(C8.D4)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8.C22, C5⋊D4, C22×D5, C8.D4, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊D4, C20.C23, D4.9D10, C5⋊(C8.D4)
►On 160 points
Generators in S160
(1 45 24 142 81)(2 82 143 17 46)(3 47 18 144 83)(4 84 137 19 48)(5 41 20 138 85)(6 86 139 21 42)(7 43 22 140 87)(8 88 141 23 44)(9 35 62 89 156)(10 157 90 63 36)(11 37 64 91 158)(12 159 92 57 38)(13 39 58 93 160)(14 153 94 59 40)(15 33 60 95 154)(16 155 96 61 34)(25 151 109 71 122)(26 123 72 110 152)(27 145 111 65 124)(28 125 66 112 146)(29 147 105 67 126)(30 127 68 106 148)(31 149 107 69 128)(32 121 70 108 150)(49 75 132 119 101)(50 102 120 133 76)(51 77 134 113 103)(52 104 114 135 78)(53 79 136 115 97)(54 98 116 129 80)(55 73 130 117 99)(56 100 118 131 74)
(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 10 27 53)(2 13 28 56)(3 16 29 51)(4 11 30 54)(5 14 31 49)(6 9 32 52)(7 12 25 55)(8 15 26 50)(17 93 112 131)(18 96 105 134)(19 91 106 129)(20 94 107 132)(21 89 108 135)(22 92 109 130)(23 95 110 133)(24 90 111 136)(33 123 102 88)(34 126 103 83)(35 121 104 86)(36 124 97 81)(37 127 98 84)(38 122 99 87)(39 125 100 82)(40 128 101 85)(41 153 149 75)(42 156 150 78)(43 159 151 73)(44 154 152 76)(45 157 145 79)(46 160 146 74)(47 155 147 77)(48 158 148 80)(57 71 117 140)(58 66 118 143)(59 69 119 138)(60 72 120 141)(61 67 113 144)(62 70 114 139)(63 65 115 142)(64 68 116 137)
(1 49 5 53)(2 56 6 52)(3 55 7 51)(4 54 8 50)(9 28 13 32)(10 27 14 31)(11 26 15 30)(12 25 16 29)(17 131 21 135)(18 130 22 134)(19 129 23 133)(20 136 24 132)(33 127 37 123)(34 126 38 122)(35 125 39 121)(36 124 40 128)(41 79 45 75)(42 78 46 74)(43 77 47 73)(44 76 48 80)(57 71 61 67)(58 70 62 66)(59 69 63 65)(60 68 64 72)(81 101 85 97)(82 100 86 104)(83 99 87 103)(84 98 88 102)(89 112 93 108)(90 111 94 107)(91 110 95 106)(92 109 96 105)(113 144 117 140)(114 143 118 139)(115 142 119 138)(116 141 120 137)(145 153 149 157)(146 160 150 156)(147 159 151 155)(148 158 152 154)
G:=sub<Sym(160)| (1,45,24,142,81)(2,82,143,17,46)(3,47,18,144,83)(4,84,137,19,48)(5,41,20,138,85)(6,86,139,21,42)(7,43,22,140,87)(8,88,141,23,44)(9,35,62,89,156)(10,157,90,63,36)(11,37,64,91,158)(12,159,92,57,38)(13,39,58,93,160)(14,153,94,59,40)(15,33,60,95,154)(16,155,96,61,34)(25,151,109,71,122)(26,123,72,110,152)(27,145,111,65,124)(28,125,66,112,146)(29,147,105,67,126)(30,127,68,106,148)(31,149,107,69,128)(32,121,70,108,150)(49,75,132,119,101)(50,102,120,133,76)(51,77,134,113,103)(52,104,114,135,78)(53,79,136,115,97)(54,98,116,129,80)(55,73,130,117,99)(56,100,118,131,74), (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), (1,10,27,53)(2,13,28,56)(3,16,29,51)(4,11,30,54)(5,14,31,49)(6,9,32,52)(7,12,25,55)(8,15,26,50)(17,93,112,131)(18,96,105,134)(19,91,106,129)(20,94,107,132)(21,89,108,135)(22,92,109,130)(23,95,110,133)(24,90,111,136)(33,123,102,88)(34,126,103,83)(35,121,104,86)(36,124,97,81)(37,127,98,84)(38,122,99,87)(39,125,100,82)(40,128,101,85)(41,153,149,75)(42,156,150,78)(43,159,151,73)(44,154,152,76)(45,157,145,79)(46,160,146,74)(47,155,147,77)(48,158,148,80)(57,71,117,140)(58,66,118,143)(59,69,119,138)(60,72,120,141)(61,67,113,144)(62,70,114,139)(63,65,115,142)(64,68,116,137), (1,49,5,53)(2,56,6,52)(3,55,7,51)(4,54,8,50)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,131,21,135)(18,130,22,134)(19,129,23,133)(20,136,24,132)(33,127,37,123)(34,126,38,122)(35,125,39,121)(36,124,40,128)(41,79,45,75)(42,78,46,74)(43,77,47,73)(44,76,48,80)(57,71,61,67)(58,70,62,66)(59,69,63,65)(60,68,64,72)(81,101,85,97)(82,100,86,104)(83,99,87,103)(84,98,88,102)(89,112,93,108)(90,111,94,107)(91,110,95,106)(92,109,96,105)(113,144,117,140)(114,143,118,139)(115,142,119,138)(116,141,120,137)(145,153,149,157)(146,160,150,156)(147,159,151,155)(148,158,152,154)>;
G:=Group( (1,45,24,142,81)(2,82,143,17,46)(3,47,18,144,83)(4,84,137,19,48)(5,41,20,138,85)(6,86,139,21,42)(7,43,22,140,87)(8,88,141,23,44)(9,35,62,89,156)(10,157,90,63,36)(11,37,64,91,158)(12,159,92,57,38)(13,39,58,93,160)(14,153,94,59,40)(15,33,60,95,154)(16,155,96,61,34)(25,151,109,71,122)(26,123,72,110,152)(27,145,111,65,124)(28,125,66,112,146)(29,147,105,67,126)(30,127,68,106,148)(31,149,107,69,128)(32,121,70,108,150)(49,75,132,119,101)(50,102,120,133,76)(51,77,134,113,103)(52,104,114,135,78)(53,79,136,115,97)(54,98,116,129,80)(55,73,130,117,99)(56,100,118,131,74), (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), (1,10,27,53)(2,13,28,56)(3,16,29,51)(4,11,30,54)(5,14,31,49)(6,9,32,52)(7,12,25,55)(8,15,26,50)(17,93,112,131)(18,96,105,134)(19,91,106,129)(20,94,107,132)(21,89,108,135)(22,92,109,130)(23,95,110,133)(24,90,111,136)(33,123,102,88)(34,126,103,83)(35,121,104,86)(36,124,97,81)(37,127,98,84)(38,122,99,87)(39,125,100,82)(40,128,101,85)(41,153,149,75)(42,156,150,78)(43,159,151,73)(44,154,152,76)(45,157,145,79)(46,160,146,74)(47,155,147,77)(48,158,148,80)(57,71,117,140)(58,66,118,143)(59,69,119,138)(60,72,120,141)(61,67,113,144)(62,70,114,139)(63,65,115,142)(64,68,116,137), (1,49,5,53)(2,56,6,52)(3,55,7,51)(4,54,8,50)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,131,21,135)(18,130,22,134)(19,129,23,133)(20,136,24,132)(33,127,37,123)(34,126,38,122)(35,125,39,121)(36,124,40,128)(41,79,45,75)(42,78,46,74)(43,77,47,73)(44,76,48,80)(57,71,61,67)(58,70,62,66)(59,69,63,65)(60,68,64,72)(81,101,85,97)(82,100,86,104)(83,99,87,103)(84,98,88,102)(89,112,93,108)(90,111,94,107)(91,110,95,106)(92,109,96,105)(113,144,117,140)(114,143,118,139)(115,142,119,138)(116,141,120,137)(145,153,149,157)(146,160,150,156)(147,159,151,155)(148,158,152,154) );
G=PermutationGroup([[(1,45,24,142,81),(2,82,143,17,46),(3,47,18,144,83),(4,84,137,19,48),(5,41,20,138,85),(6,86,139,21,42),(7,43,22,140,87),(8,88,141,23,44),(9,35,62,89,156),(10,157,90,63,36),(11,37,64,91,158),(12,159,92,57,38),(13,39,58,93,160),(14,153,94,59,40),(15,33,60,95,154),(16,155,96,61,34),(25,151,109,71,122),(26,123,72,110,152),(27,145,111,65,124),(28,125,66,112,146),(29,147,105,67,126),(30,127,68,106,148),(31,149,107,69,128),(32,121,70,108,150),(49,75,132,119,101),(50,102,120,133,76),(51,77,134,113,103),(52,104,114,135,78),(53,79,136,115,97),(54,98,116,129,80),(55,73,130,117,99),(56,100,118,131,74)], [(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)], [(1,10,27,53),(2,13,28,56),(3,16,29,51),(4,11,30,54),(5,14,31,49),(6,9,32,52),(7,12,25,55),(8,15,26,50),(17,93,112,131),(18,96,105,134),(19,91,106,129),(20,94,107,132),(21,89,108,135),(22,92,109,130),(23,95,110,133),(24,90,111,136),(33,123,102,88),(34,126,103,83),(35,121,104,86),(36,124,97,81),(37,127,98,84),(38,122,99,87),(39,125,100,82),(40,128,101,85),(41,153,149,75),(42,156,150,78),(43,159,151,73),(44,154,152,76),(45,157,145,79),(46,160,146,74),(47,155,147,77),(48,158,148,80),(57,71,117,140),(58,66,118,143),(59,69,119,138),(60,72,120,141),(61,67,113,144),(62,70,114,139),(63,65,115,142),(64,68,116,137)], [(1,49,5,53),(2,56,6,52),(3,55,7,51),(4,54,8,50),(9,28,13,32),(10,27,14,31),(11,26,15,30),(12,25,16,29),(17,131,21,135),(18,130,22,134),(19,129,23,133),(20,136,24,132),(33,127,37,123),(34,126,38,122),(35,125,39,121),(36,124,40,128),(41,79,45,75),(42,78,46,74),(43,77,47,73),(44,76,48,80),(57,71,61,67),(58,70,62,66),(59,69,63,65),(60,68,64,72),(81,101,85,97),(82,100,86,104),(83,99,87,103),(84,98,88,102),(89,112,93,108),(90,111,94,107),(91,110,95,106),(92,109,96,105),(113,144,117,140),(114,143,118,139),(115,142,119,138),(116,141,120,137),(145,153,149,157),(146,160,150,156),(147,159,151,155),(148,158,152,154)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 8 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | D4×D5 | D4⋊2D5 | C20.C23 | D4.9D10 |
| kernel | C5⋊(C8.D4) | C20.Q8 | C10.Q16 | Q8⋊Dic5 | C2×C4.Dic5 | C20.48D4 | C2×C5⋊Q16 | C5×C22⋊Q8 | C5⋊2C8 | C2×C20 | C22×C10 | C22⋊Q8 | C20 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C23 | C10 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C5⋊(C8.D4) ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 34 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 22 | 23 | 38 |
| 0 | 0 | 0 | 32 | 0 | 18 |
| 0 | 0 | 13 | 30 | 32 | 19 |
| 0 | 0 | 21 | 28 | 0 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 19 | 23 |
| 0 | 0 | 32 | 30 | 18 | 22 |
| 0 | 0 | 11 | 9 | 30 | 32 |
| 0 | 0 | 32 | 30 | 9 | 11 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 19 | 23 |
| 0 | 0 | 32 | 30 | 18 | 22 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 0 | 0 | 0 | 0 | 9 | 11 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,1,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,13,21,0,0,22,32,30,28,0,0,23,0,32,0,0,0,38,18,19,9],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,11,32,11,32,0,0,9,30,9,30,0,0,19,18,30,9,0,0,23,22,32,11],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,19,18,30,9,0,0,23,22,32,11] >;
C5⋊(C8.D4) in GAP, Magma, Sage, TeX
C_5\rtimes (C_8.D_4)
% in TeX
G:=Group("C5:(C8.D4)"); // GroupNames label
G:=SmallGroup(320,679);
// by ID
G=gap.SmallGroup(320,679);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,555,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=b^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations