direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8.18D4, C40.98D4, (C2×C10)⋊4Q16, C2.D8⋊3C10, C8.18(C5×D4), (C2×Q16)⋊4C10, C4.59(D4×C10), C2.6(C10×Q16), C22⋊1(C5×Q16), Q8⋊C4⋊2C10, (C10×Q16)⋊18C2, (C2×C20).365D4, C20.466(C2×D4), (C22×C8).9C10, C10.53(C2×Q16), C23.27(C5×D4), C22⋊Q8.3C10, (C22×C40).27C2, C22.91(D4×C10), C20.264(C4○D4), C10.125(C4○D8), (C2×C40).365C22, (C2×C20).926C23, (C22×C10).131D4, C10.150(C4⋊D4), (Q8×C10).164C22, (C22×C20).593C22, C4.9(C5×C4○D4), C4⋊C4.7(C2×C10), (C5×C2.D8)⋊18C2, C2.12(C5×C4○D8), (C2×C4).55(C5×D4), (C2×C8).78(C2×C10), (C5×Q8⋊C4)⋊2C2, C2.19(C5×C4⋊D4), (C2×Q8).8(C2×C10), (C2×C10).647(C2×D4), (C5×C22⋊Q8).13C2, (C5×C4⋊C4).229C22, (C2×C4).101(C22×C10), (C22×C4).122(C2×C10), SmallGroup(320,968)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.18D4
G = < a,b,c,d | a5=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b4c-1 >
Subgroups: 186 in 114 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C40, C40, C2×C20, C2×C20, C5×Q8, C22×C10, C8.18D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×Q16, C22×C20, Q8×C10, C5×Q8⋊C4, C5×C2.D8, C5×C22⋊Q8, C22×C40, C10×Q16, C5×C8.18D4
Quotients: C1, C2, C22, C5, D4, C23, C10, Q16, C2×D4, C4○D4, C2×C10, C4⋊D4, C2×Q16, C4○D8, C5×D4, C22×C10, C8.18D4, C5×Q16, D4×C10, C5×C4○D4, C5×C4⋊D4, C10×Q16, C5×C4○D8, C5×C8.18D4
►On 160 points
Generators in S160
(1 85 75 33 67)(2 86 76 34 68)(3 87 77 35 69)(4 88 78 36 70)(5 81 79 37 71)(6 82 80 38 72)(7 83 73 39 65)(8 84 74 40 66)(9 27 18 57 51)(10 28 19 58 52)(11 29 20 59 53)(12 30 21 60 54)(13 31 22 61 55)(14 32 23 62 56)(15 25 24 63 49)(16 26 17 64 50)(41 154 116 146 108)(42 155 117 147 109)(43 156 118 148 110)(44 157 119 149 111)(45 158 120 150 112)(46 159 113 151 105)(47 160 114 152 106)(48 153 115 145 107)(89 127 143 97 135)(90 128 144 98 136)(91 121 137 99 129)(92 122 138 100 130)(93 123 139 101 131)(94 124 140 102 132)(95 125 141 103 133)(96 126 142 104 134)
(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)
(1 126 11 107)(2 125 12 106)(3 124 13 105)(4 123 14 112)(5 122 15 111)(6 121 16 110)(7 128 9 109)(8 127 10 108)(17 156 80 99)(18 155 73 98)(19 154 74 97)(20 153 75 104)(21 160 76 103)(22 159 77 102)(23 158 78 101)(24 157 79 100)(25 44 81 138)(26 43 82 137)(27 42 83 144)(28 41 84 143)(29 48 85 142)(30 47 86 141)(31 46 87 140)(32 45 88 139)(33 134 59 115)(34 133 60 114)(35 132 61 113)(36 131 62 120)(37 130 63 119)(38 129 64 118)(39 136 57 117)(40 135 58 116)(49 149 71 92)(50 148 72 91)(51 147 65 90)(52 146 66 89)(53 145 67 96)(54 152 68 95)(55 151 69 94)(56 150 70 93)
(1 107 5 111)(2 106 6 110)(3 105 7 109)(4 112 8 108)(9 128 13 124)(10 127 14 123)(11 126 15 122)(12 125 16 121)(17 99 21 103)(18 98 22 102)(19 97 23 101)(20 104 24 100)(25 138 29 142)(26 137 30 141)(27 144 31 140)(28 143 32 139)(33 115 37 119)(34 114 38 118)(35 113 39 117)(36 120 40 116)(41 88 45 84)(42 87 46 83)(43 86 47 82)(44 85 48 81)(49 92 53 96)(50 91 54 95)(51 90 55 94)(52 89 56 93)(57 136 61 132)(58 135 62 131)(59 134 63 130)(60 133 64 129)(65 147 69 151)(66 146 70 150)(67 145 71 149)(68 152 72 148)(73 155 77 159)(74 154 78 158)(75 153 79 157)(76 160 80 156)
G:=sub<Sym(160)| (1,85,75,33,67)(2,86,76,34,68)(3,87,77,35,69)(4,88,78,36,70)(5,81,79,37,71)(6,82,80,38,72)(7,83,73,39,65)(8,84,74,40,66)(9,27,18,57,51)(10,28,19,58,52)(11,29,20,59,53)(12,30,21,60,54)(13,31,22,61,55)(14,32,23,62,56)(15,25,24,63,49)(16,26,17,64,50)(41,154,116,146,108)(42,155,117,147,109)(43,156,118,148,110)(44,157,119,149,111)(45,158,120,150,112)(46,159,113,151,105)(47,160,114,152,106)(48,153,115,145,107)(89,127,143,97,135)(90,128,144,98,136)(91,121,137,99,129)(92,122,138,100,130)(93,123,139,101,131)(94,124,140,102,132)(95,125,141,103,133)(96,126,142,104,134), (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), (1,126,11,107)(2,125,12,106)(3,124,13,105)(4,123,14,112)(5,122,15,111)(6,121,16,110)(7,128,9,109)(8,127,10,108)(17,156,80,99)(18,155,73,98)(19,154,74,97)(20,153,75,104)(21,160,76,103)(22,159,77,102)(23,158,78,101)(24,157,79,100)(25,44,81,138)(26,43,82,137)(27,42,83,144)(28,41,84,143)(29,48,85,142)(30,47,86,141)(31,46,87,140)(32,45,88,139)(33,134,59,115)(34,133,60,114)(35,132,61,113)(36,131,62,120)(37,130,63,119)(38,129,64,118)(39,136,57,117)(40,135,58,116)(49,149,71,92)(50,148,72,91)(51,147,65,90)(52,146,66,89)(53,145,67,96)(54,152,68,95)(55,151,69,94)(56,150,70,93), (1,107,5,111)(2,106,6,110)(3,105,7,109)(4,112,8,108)(9,128,13,124)(10,127,14,123)(11,126,15,122)(12,125,16,121)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,138,29,142)(26,137,30,141)(27,144,31,140)(28,143,32,139)(33,115,37,119)(34,114,38,118)(35,113,39,117)(36,120,40,116)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,92,53,96)(50,91,54,95)(51,90,55,94)(52,89,56,93)(57,136,61,132)(58,135,62,131)(59,134,63,130)(60,133,64,129)(65,147,69,151)(66,146,70,150)(67,145,71,149)(68,152,72,148)(73,155,77,159)(74,154,78,158)(75,153,79,157)(76,160,80,156)>;
G:=Group( (1,85,75,33,67)(2,86,76,34,68)(3,87,77,35,69)(4,88,78,36,70)(5,81,79,37,71)(6,82,80,38,72)(7,83,73,39,65)(8,84,74,40,66)(9,27,18,57,51)(10,28,19,58,52)(11,29,20,59,53)(12,30,21,60,54)(13,31,22,61,55)(14,32,23,62,56)(15,25,24,63,49)(16,26,17,64,50)(41,154,116,146,108)(42,155,117,147,109)(43,156,118,148,110)(44,157,119,149,111)(45,158,120,150,112)(46,159,113,151,105)(47,160,114,152,106)(48,153,115,145,107)(89,127,143,97,135)(90,128,144,98,136)(91,121,137,99,129)(92,122,138,100,130)(93,123,139,101,131)(94,124,140,102,132)(95,125,141,103,133)(96,126,142,104,134), (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), (1,126,11,107)(2,125,12,106)(3,124,13,105)(4,123,14,112)(5,122,15,111)(6,121,16,110)(7,128,9,109)(8,127,10,108)(17,156,80,99)(18,155,73,98)(19,154,74,97)(20,153,75,104)(21,160,76,103)(22,159,77,102)(23,158,78,101)(24,157,79,100)(25,44,81,138)(26,43,82,137)(27,42,83,144)(28,41,84,143)(29,48,85,142)(30,47,86,141)(31,46,87,140)(32,45,88,139)(33,134,59,115)(34,133,60,114)(35,132,61,113)(36,131,62,120)(37,130,63,119)(38,129,64,118)(39,136,57,117)(40,135,58,116)(49,149,71,92)(50,148,72,91)(51,147,65,90)(52,146,66,89)(53,145,67,96)(54,152,68,95)(55,151,69,94)(56,150,70,93), (1,107,5,111)(2,106,6,110)(3,105,7,109)(4,112,8,108)(9,128,13,124)(10,127,14,123)(11,126,15,122)(12,125,16,121)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,138,29,142)(26,137,30,141)(27,144,31,140)(28,143,32,139)(33,115,37,119)(34,114,38,118)(35,113,39,117)(36,120,40,116)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,92,53,96)(50,91,54,95)(51,90,55,94)(52,89,56,93)(57,136,61,132)(58,135,62,131)(59,134,63,130)(60,133,64,129)(65,147,69,151)(66,146,70,150)(67,145,71,149)(68,152,72,148)(73,155,77,159)(74,154,78,158)(75,153,79,157)(76,160,80,156) );
G=PermutationGroup([[(1,85,75,33,67),(2,86,76,34,68),(3,87,77,35,69),(4,88,78,36,70),(5,81,79,37,71),(6,82,80,38,72),(7,83,73,39,65),(8,84,74,40,66),(9,27,18,57,51),(10,28,19,58,52),(11,29,20,59,53),(12,30,21,60,54),(13,31,22,61,55),(14,32,23,62,56),(15,25,24,63,49),(16,26,17,64,50),(41,154,116,146,108),(42,155,117,147,109),(43,156,118,148,110),(44,157,119,149,111),(45,158,120,150,112),(46,159,113,151,105),(47,160,114,152,106),(48,153,115,145,107),(89,127,143,97,135),(90,128,144,98,136),(91,121,137,99,129),(92,122,138,100,130),(93,123,139,101,131),(94,124,140,102,132),(95,125,141,103,133),(96,126,142,104,134)], [(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)], [(1,126,11,107),(2,125,12,106),(3,124,13,105),(4,123,14,112),(5,122,15,111),(6,121,16,110),(7,128,9,109),(8,127,10,108),(17,156,80,99),(18,155,73,98),(19,154,74,97),(20,153,75,104),(21,160,76,103),(22,159,77,102),(23,158,78,101),(24,157,79,100),(25,44,81,138),(26,43,82,137),(27,42,83,144),(28,41,84,143),(29,48,85,142),(30,47,86,141),(31,46,87,140),(32,45,88,139),(33,134,59,115),(34,133,60,114),(35,132,61,113),(36,131,62,120),(37,130,63,119),(38,129,64,118),(39,136,57,117),(40,135,58,116),(49,149,71,92),(50,148,72,91),(51,147,65,90),(52,146,66,89),(53,145,67,96),(54,152,68,95),(55,151,69,94),(56,150,70,93)], [(1,107,5,111),(2,106,6,110),(3,105,7,109),(4,112,8,108),(9,128,13,124),(10,127,14,123),(11,126,15,122),(12,125,16,121),(17,99,21,103),(18,98,22,102),(19,97,23,101),(20,104,24,100),(25,138,29,142),(26,137,30,141),(27,144,31,140),(28,143,32,139),(33,115,37,119),(34,114,38,118),(35,113,39,117),(36,120,40,116),(41,88,45,84),(42,87,46,83),(43,86,47,82),(44,85,48,81),(49,92,53,96),(50,91,54,95),(51,90,55,94),(52,89,56,93),(57,136,61,132),(58,135,62,131),(59,134,63,130),(60,133,64,129),(65,147,69,151),(66,146,70,150),(67,145,71,149),(68,152,72,148),(73,155,77,159),(74,154,78,158),(75,153,79,157),(76,160,80,156)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | C4○D4 | Q16 | C4○D8 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D4 | C5×Q16 | C5×C4○D8 |
| kernel | C5×C8.18D4 | C5×Q8⋊C4 | C5×C2.D8 | C5×C22⋊Q8 | C22×C40 | C10×Q16 | C8.18D4 | Q8⋊C4 | C2.D8 | C22⋊Q8 | C22×C8 | C2×Q16 | C40 | C2×C20 | C22×C10 | C20 | C2×C10 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 8 | 4 | 8 | 4 | 4 | 2 | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C5×C8.18D4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 26 | 38 |
| 40 | 40 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 40 |
| 40 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,27,26,0,0,0,38],[40,2,0,0,40,1,0,0,0,0,1,0,0,0,2,40],[40,0,0,0,40,1,0,0,0,0,40,1,0,0,39,1] >;
C5×C8.18D4 in GAP, Magma, Sage, TeX
C_5\times C_8._{18}D_4 % in TeX
G:=Group("C5xC8.18D4"); // GroupNames label
G:=SmallGroup(320,968);
// by ID
G=gap.SmallGroup(320,968);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations