direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C10×SD32, C20.45D8, C40.73D4, C80⋊13C22, C40.73C23, C4.7(C5×D8), (C2×C16)⋊7C10, (C2×C80)⋊17C2, C16⋊3(C2×C10), C8.10(C5×D4), C4.8(D4×C10), Q16⋊1(C2×C10), (C2×Q16)⋊6C10, (C2×D8).4C10, D8.1(C2×C10), (C2×C10).56D8, C2.13(C10×D8), C10.85(C2×D8), (C10×Q16)⋊20C2, (C10×D8).11C2, C20.315(C2×D4), (C2×C20).427D4, C8.4(C22×C10), C22.15(C5×D8), (C5×Q16)⋊15C22, (C5×D8).11C22, (C2×C40).427C22, (C2×C4).83(C5×D4), (C2×C8).85(C2×C10), SmallGroup(320,1007)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 210 in 90 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C10, C10 [×2], C10 [×2], C16 [×2], C2×C8, D8 [×2], D8, Q16 [×2], Q16, C2×D4, C2×Q8, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C2×C16, SD32 [×4], C2×D8, C2×Q16, C40 [×2], C2×C20, C2×C20, C5×D4 [×3], C5×Q8 [×3], C22×C10, C2×SD32, C80 [×2], C2×C40, C5×D8 [×2], C5×D8, C5×Q16 [×2], C5×Q16, D4×C10, Q8×C10, C2×C80, C5×SD32 [×4], C10×D8, C10×Q16, C10×SD32
Quotients:
C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], SD32 [×2], C2×D8, C5×D4 [×2], C22×C10, C2×SD32, C5×D8 [×2], D4×C10, C5×SD32 [×2], C10×D8, C10×SD32
Generators and relations
G = < a,b,c | a10=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
►On 160 points
Generators in S160
(1 41 120 29 60 72 102 94 130 146)(2 42 121 30 61 73 103 95 131 147)(3 43 122 31 62 74 104 96 132 148)(4 44 123 32 63 75 105 81 133 149)(5 45 124 17 64 76 106 82 134 150)(6 46 125 18 49 77 107 83 135 151)(7 47 126 19 50 78 108 84 136 152)(8 48 127 20 51 79 109 85 137 153)(9 33 128 21 52 80 110 86 138 154)(10 34 113 22 53 65 111 87 139 155)(11 35 114 23 54 66 112 88 140 156)(12 36 115 24 55 67 97 89 141 157)(13 37 116 25 56 68 98 90 142 158)(14 38 117 26 57 69 99 91 143 159)(15 39 118 27 58 70 100 92 144 160)(16 40 119 28 59 71 101 93 129 145)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(34 40)(35 47)(36 38)(37 45)(39 43)(42 48)(44 46)(49 63)(50 54)(51 61)(53 59)(55 57)(56 64)(58 62)(65 71)(66 78)(67 69)(68 76)(70 74)(73 79)(75 77)(81 83)(82 90)(84 88)(85 95)(87 93)(89 91)(92 96)(97 99)(98 106)(100 104)(101 111)(103 109)(105 107)(108 112)(113 119)(114 126)(115 117)(116 124)(118 122)(121 127)(123 125)(129 139)(131 137)(132 144)(133 135)(134 142)(136 140)(141 143)(145 155)(147 153)(148 160)(149 151)(150 158)(152 156)(157 159)
G:=sub<Sym(160)| (1,41,120,29,60,72,102,94,130,146)(2,42,121,30,61,73,103,95,131,147)(3,43,122,31,62,74,104,96,132,148)(4,44,123,32,63,75,105,81,133,149)(5,45,124,17,64,76,106,82,134,150)(6,46,125,18,49,77,107,83,135,151)(7,47,126,19,50,78,108,84,136,152)(8,48,127,20,51,79,109,85,137,153)(9,33,128,21,52,80,110,86,138,154)(10,34,113,22,53,65,111,87,139,155)(11,35,114,23,54,66,112,88,140,156)(12,36,115,24,55,67,97,89,141,157)(13,37,116,25,56,68,98,90,142,158)(14,38,117,26,57,69,99,91,143,159)(15,39,118,27,58,70,100,92,144,160)(16,40,119,28,59,71,101,93,129,145), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(34,40)(35,47)(36,38)(37,45)(39,43)(42,48)(44,46)(49,63)(50,54)(51,61)(53,59)(55,57)(56,64)(58,62)(65,71)(66,78)(67,69)(68,76)(70,74)(73,79)(75,77)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112)(113,119)(114,126)(115,117)(116,124)(118,122)(121,127)(123,125)(129,139)(131,137)(132,144)(133,135)(134,142)(136,140)(141,143)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159)>;
G:=Group( (1,41,120,29,60,72,102,94,130,146)(2,42,121,30,61,73,103,95,131,147)(3,43,122,31,62,74,104,96,132,148)(4,44,123,32,63,75,105,81,133,149)(5,45,124,17,64,76,106,82,134,150)(6,46,125,18,49,77,107,83,135,151)(7,47,126,19,50,78,108,84,136,152)(8,48,127,20,51,79,109,85,137,153)(9,33,128,21,52,80,110,86,138,154)(10,34,113,22,53,65,111,87,139,155)(11,35,114,23,54,66,112,88,140,156)(12,36,115,24,55,67,97,89,141,157)(13,37,116,25,56,68,98,90,142,158)(14,38,117,26,57,69,99,91,143,159)(15,39,118,27,58,70,100,92,144,160)(16,40,119,28,59,71,101,93,129,145), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(34,40)(35,47)(36,38)(37,45)(39,43)(42,48)(44,46)(49,63)(50,54)(51,61)(53,59)(55,57)(56,64)(58,62)(65,71)(66,78)(67,69)(68,76)(70,74)(73,79)(75,77)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112)(113,119)(114,126)(115,117)(116,124)(118,122)(121,127)(123,125)(129,139)(131,137)(132,144)(133,135)(134,142)(136,140)(141,143)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159) );
G=PermutationGroup([(1,41,120,29,60,72,102,94,130,146),(2,42,121,30,61,73,103,95,131,147),(3,43,122,31,62,74,104,96,132,148),(4,44,123,32,63,75,105,81,133,149),(5,45,124,17,64,76,106,82,134,150),(6,46,125,18,49,77,107,83,135,151),(7,47,126,19,50,78,108,84,136,152),(8,48,127,20,51,79,109,85,137,153),(9,33,128,21,52,80,110,86,138,154),(10,34,113,22,53,65,111,87,139,155),(11,35,114,23,54,66,112,88,140,156),(12,36,115,24,55,67,97,89,141,157),(13,37,116,25,56,68,98,90,142,158),(14,38,117,26,57,69,99,91,143,159),(15,39,118,27,58,70,100,92,144,160),(16,40,119,28,59,71,101,93,129,145)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(34,40),(35,47),(36,38),(37,45),(39,43),(42,48),(44,46),(49,63),(50,54),(51,61),(53,59),(55,57),(56,64),(58,62),(65,71),(66,78),(67,69),(68,76),(70,74),(73,79),(75,77),(81,83),(82,90),(84,88),(85,95),(87,93),(89,91),(92,96),(97,99),(98,106),(100,104),(101,111),(103,109),(105,107),(108,112),(113,119),(114,126),(115,117),(116,124),(118,122),(121,127),(123,125),(129,139),(131,137),(132,144),(133,135),(134,142),(136,140),(141,143),(145,155),(147,153),(148,160),(149,151),(150,158),(152,156),(157,159)])
Matrix representation ►G ⊆ GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 87 | 0 |
| 0 | 0 | 87 |
| 1 | 0 | 0 |
| 0 | 103 | 41 |
| 0 | 200 | 103 |
| 240 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 240 |
G:=sub<GL(3,GF(241))| [240,0,0,0,87,0,0,0,87],[1,0,0,0,103,200,0,41,103],[240,0,0,0,1,0,0,0,240] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 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 | C5 | C10 | C10 | C10 | C10 | D4 | D4 | D8 | D8 | SD32 | C5×D4 | C5×D4 | C5×D8 | C5×D8 | C5×SD32 |
| kernel | C10×SD32 | C2×C80 | C5×SD32 | C10×D8 | C10×Q16 | C2×SD32 | C2×C16 | SD32 | C2×D8 | C2×Q16 | C40 | C2×C20 | C20 | C2×C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 1 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times SD_{32} % in TeX
G:=Group("C10xSD32"); // GroupNames label
G:=SmallGroup(320,1007);
// by ID
G=gap.SmallGroup(320,1007);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,4204,2111,242,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c|a^10=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations
×
⋊
ℤ
𝔽
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