direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C4○D16, D16⋊3C10, Q32⋊3C10, C20.71D8, C40.76D4, SD32⋊3C10, C80.23C22, C40.75C23, (C2×C16)⋊6C10, (C2×C80)⋊13C2, C4○D8⋊1C10, (C5×D16)⋊7C2, (C5×Q32)⋊7C2, C8.13(C5×D4), C4.20(C5×D8), C16.6(C2×C10), (C5×SD32)⋊7C2, D8.2(C2×C10), C10.87(C2×D8), C4.10(D4×C10), C2.15(C10×D8), (C2×C10).12D8, C22.1(C5×D8), C20.317(C2×D4), (C2×C20).429D4, C8.6(C22×C10), Q16.2(C2×C10), (C5×D8).12C22, (C2×C40).434C22, (C5×Q16).14C22, (C5×C4○D8)⋊8C2, (C2×C4).85(C5×D4), (C2×C8).91(C2×C10), SmallGroup(320,1009)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C10, C10 [×3], C16 [×2], C2×C8, D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C16, D16, SD32 [×2], Q32, C4○D8 [×2], C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×4], C5×Q8 [×2], C4○D16, C80 [×2], C2×C40, C5×D8 [×2], C5×SD16 [×2], C5×Q16 [×2], C5×C4○D4 [×2], C2×C80, C5×D16, C5×SD32 [×2], C5×Q32, C5×C4○D8 [×2], C5×C4○D16
Quotients:
C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], C2×D8, C5×D4 [×2], C22×C10, C4○D16, C5×D8 [×2], D4×C10, C10×D8, C5×C4○D16
Generators and relations
G = < a,b,c,d | a5=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >
►On 160 points
Generators in S160
(1 127 137 110 95)(2 128 138 111 96)(3 113 139 112 81)(4 114 140 97 82)(5 115 141 98 83)(6 116 142 99 84)(7 117 143 100 85)(8 118 144 101 86)(9 119 129 102 87)(10 120 130 103 88)(11 121 131 104 89)(12 122 132 105 90)(13 123 133 106 91)(14 124 134 107 92)(15 125 135 108 93)(16 126 136 109 94)(17 150 58 66 33)(18 151 59 67 34)(19 152 60 68 35)(20 153 61 69 36)(21 154 62 70 37)(22 155 63 71 38)(23 156 64 72 39)(24 157 49 73 40)(25 158 50 74 41)(26 159 51 75 42)(27 160 52 76 43)(28 145 53 77 44)(29 146 54 78 45)(30 147 55 79 46)(31 148 56 80 47)(32 149 57 65 48)
(1 36 9 44)(2 37 10 45)(3 38 11 46)(4 39 12 47)(5 40 13 48)(6 41 14 33)(7 42 15 34)(8 43 16 35)(17 116 25 124)(18 117 26 125)(19 118 27 126)(20 119 28 127)(21 120 29 128)(22 121 30 113)(23 122 31 114)(24 123 32 115)(49 106 57 98)(50 107 58 99)(51 108 59 100)(52 109 60 101)(53 110 61 102)(54 111 62 103)(55 112 63 104)(56 97 64 105)(65 83 73 91)(66 84 74 92)(67 85 75 93)(68 86 76 94)(69 87 77 95)(70 88 78 96)(71 89 79 81)(72 90 80 82)(129 145 137 153)(130 146 138 154)(131 147 139 155)(132 148 140 156)(133 149 141 157)(134 150 142 158)(135 151 143 159)(136 152 144 160)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 22)(18 21)(19 20)(23 32)(24 31)(25 30)(26 29)(27 28)(33 38)(34 37)(35 36)(39 48)(40 47)(41 46)(42 45)(43 44)(49 56)(50 55)(51 54)(52 53)(57 64)(58 63)(59 62)(60 61)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 96)(94 95)(97 106)(98 105)(99 104)(100 103)(101 102)(107 112)(108 111)(109 110)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(125 128)(126 127)(129 144)(130 143)(131 142)(132 141)(133 140)(134 139)(135 138)(136 137)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)
G:=sub<Sym(160)| (1,127,137,110,95)(2,128,138,111,96)(3,113,139,112,81)(4,114,140,97,82)(5,115,141,98,83)(6,116,142,99,84)(7,117,143,100,85)(8,118,144,101,86)(9,119,129,102,87)(10,120,130,103,88)(11,121,131,104,89)(12,122,132,105,90)(13,123,133,106,91)(14,124,134,107,92)(15,125,135,108,93)(16,126,136,109,94)(17,150,58,66,33)(18,151,59,67,34)(19,152,60,68,35)(20,153,61,69,36)(21,154,62,70,37)(22,155,63,71,38)(23,156,64,72,39)(24,157,49,73,40)(25,158,50,74,41)(26,159,51,75,42)(27,160,52,76,43)(28,145,53,77,44)(29,146,54,78,45)(30,147,55,79,46)(31,148,56,80,47)(32,149,57,65,48), (1,36,9,44)(2,37,10,45)(3,38,11,46)(4,39,12,47)(5,40,13,48)(6,41,14,33)(7,42,15,34)(8,43,16,35)(17,116,25,124)(18,117,26,125)(19,118,27,126)(20,119,28,127)(21,120,29,128)(22,121,30,113)(23,122,31,114)(24,123,32,115)(49,106,57,98)(50,107,58,99)(51,108,59,100)(52,109,60,101)(53,110,61,102)(54,111,62,103)(55,112,63,104)(56,97,64,105)(65,83,73,91)(66,84,74,92)(67,85,75,93)(68,86,76,94)(69,87,77,95)(70,88,78,96)(71,89,79,81)(72,90,80,82)(129,145,137,153)(130,146,138,154)(131,147,139,155)(132,148,140,156)(133,149,141,157)(134,150,142,158)(135,151,143,159)(136,152,144,160), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,38)(34,37)(35,36)(39,48)(40,47)(41,46)(42,45)(43,44)(49,56)(50,55)(51,54)(52,53)(57,64)(58,63)(59,62)(60,61)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95)(97,106)(98,105)(99,104)(100,103)(101,102)(107,112)(108,111)(109,110)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(125,128)(126,127)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)>;
G:=Group( (1,127,137,110,95)(2,128,138,111,96)(3,113,139,112,81)(4,114,140,97,82)(5,115,141,98,83)(6,116,142,99,84)(7,117,143,100,85)(8,118,144,101,86)(9,119,129,102,87)(10,120,130,103,88)(11,121,131,104,89)(12,122,132,105,90)(13,123,133,106,91)(14,124,134,107,92)(15,125,135,108,93)(16,126,136,109,94)(17,150,58,66,33)(18,151,59,67,34)(19,152,60,68,35)(20,153,61,69,36)(21,154,62,70,37)(22,155,63,71,38)(23,156,64,72,39)(24,157,49,73,40)(25,158,50,74,41)(26,159,51,75,42)(27,160,52,76,43)(28,145,53,77,44)(29,146,54,78,45)(30,147,55,79,46)(31,148,56,80,47)(32,149,57,65,48), (1,36,9,44)(2,37,10,45)(3,38,11,46)(4,39,12,47)(5,40,13,48)(6,41,14,33)(7,42,15,34)(8,43,16,35)(17,116,25,124)(18,117,26,125)(19,118,27,126)(20,119,28,127)(21,120,29,128)(22,121,30,113)(23,122,31,114)(24,123,32,115)(49,106,57,98)(50,107,58,99)(51,108,59,100)(52,109,60,101)(53,110,61,102)(54,111,62,103)(55,112,63,104)(56,97,64,105)(65,83,73,91)(66,84,74,92)(67,85,75,93)(68,86,76,94)(69,87,77,95)(70,88,78,96)(71,89,79,81)(72,90,80,82)(129,145,137,153)(130,146,138,154)(131,147,139,155)(132,148,140,156)(133,149,141,157)(134,150,142,158)(135,151,143,159)(136,152,144,160), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,38)(34,37)(35,36)(39,48)(40,47)(41,46)(42,45)(43,44)(49,56)(50,55)(51,54)(52,53)(57,64)(58,63)(59,62)(60,61)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,96)(94,95)(97,106)(98,105)(99,104)(100,103)(101,102)(107,112)(108,111)(109,110)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(125,128)(126,127)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153) );
G=PermutationGroup([(1,127,137,110,95),(2,128,138,111,96),(3,113,139,112,81),(4,114,140,97,82),(5,115,141,98,83),(6,116,142,99,84),(7,117,143,100,85),(8,118,144,101,86),(9,119,129,102,87),(10,120,130,103,88),(11,121,131,104,89),(12,122,132,105,90),(13,123,133,106,91),(14,124,134,107,92),(15,125,135,108,93),(16,126,136,109,94),(17,150,58,66,33),(18,151,59,67,34),(19,152,60,68,35),(20,153,61,69,36),(21,154,62,70,37),(22,155,63,71,38),(23,156,64,72,39),(24,157,49,73,40),(25,158,50,74,41),(26,159,51,75,42),(27,160,52,76,43),(28,145,53,77,44),(29,146,54,78,45),(30,147,55,79,46),(31,148,56,80,47),(32,149,57,65,48)], [(1,36,9,44),(2,37,10,45),(3,38,11,46),(4,39,12,47),(5,40,13,48),(6,41,14,33),(7,42,15,34),(8,43,16,35),(17,116,25,124),(18,117,26,125),(19,118,27,126),(20,119,28,127),(21,120,29,128),(22,121,30,113),(23,122,31,114),(24,123,32,115),(49,106,57,98),(50,107,58,99),(51,108,59,100),(52,109,60,101),(53,110,61,102),(54,111,62,103),(55,112,63,104),(56,97,64,105),(65,83,73,91),(66,84,74,92),(67,85,75,93),(68,86,76,94),(69,87,77,95),(70,88,78,96),(71,89,79,81),(72,90,80,82),(129,145,137,153),(130,146,138,154),(131,147,139,155),(132,148,140,156),(133,149,141,157),(134,150,142,158),(135,151,143,159),(136,152,144,160)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,22),(18,21),(19,20),(23,32),(24,31),(25,30),(26,29),(27,28),(33,38),(34,37),(35,36),(39,48),(40,47),(41,46),(42,45),(43,44),(49,56),(50,55),(51,54),(52,53),(57,64),(58,63),(59,62),(60,61),(65,72),(66,71),(67,70),(68,69),(73,80),(74,79),(75,78),(76,77),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,96),(94,95),(97,106),(98,105),(99,104),(100,103),(101,102),(107,112),(108,111),(109,110),(113,124),(114,123),(115,122),(116,121),(117,120),(118,119),(125,128),(126,127),(129,144),(130,143),(131,142),(132,141),(133,140),(134,139),(135,138),(136,137),(145,160),(146,159),(147,158),(148,157),(149,156),(150,155),(151,154),(152,153)])
Matrix representation ►G ⊆ GL2(𝔽241) generated by
| 91 | 0 |
| 0 | 91 |
| 177 | 0 |
| 0 | 177 |
| 183 | 187 |
| 27 | 129 |
| 183 | 187 |
| 156 | 58 |
G:=sub<GL(2,GF(241))| [91,0,0,91],[177,0,0,177],[183,27,187,129],[183,156,187,58] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 16A | ··· | 16H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | C5×D4 | C5×D4 | C4○D16 | C5×D8 | C5×D8 | C5×C4○D16 |
| kernel | C5×C4○D16 | C2×C80 | C5×D16 | C5×SD32 | C5×Q32 | C5×C4○D8 | C4○D16 | C2×C16 | D16 | SD32 | Q32 | C4○D8 | C40 | C2×C20 | C20 | C2×C10 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_5\times C_4\circ D_{16} % in TeX
G:=Group("C5xC4oD16"); // GroupNames label
G:=SmallGroup(320,1009);
// by ID
G=gap.SmallGroup(320,1009);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,856,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀