direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×M5(2), C80⋊14C22, C23.3C40, C40.82C23, (C2×C16)⋊8C10, (C2×C80)⋊18C2, C16⋊4(C2×C10), (C2×C4).6C40, C8.20(C2×C20), C20.84(C2×C8), (C2×C40).57C4, C4.10(C2×C40), (C2×C20).24C8, (C2×C8).14C20, C40.130(C2×C4), C22.6(C2×C40), (C22×C10).8C8, C2.6(C22×C40), C10.59(C22×C8), (C22×C8).16C10, (C22×C4).17C20, C4.35(C22×C20), C8.15(C22×C10), (C22×C20).66C4, (C22×C40).34C2, C20.252(C22×C4), (C2×C40).448C22, (C2×C10).52(C2×C8), (C2×C4).85(C2×C20), (C2×C8).102(C2×C10), (C2×C20).513(C2×C4), SmallGroup(320,1004)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 98 in 90 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C5, C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×4], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C16 [×2], M5(2) [×4], C22×C8, C40 [×2], C40 [×2], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×M5(2), C80 [×4], C2×C40 [×2], C2×C40 [×4], C22×C20, C2×C80 [×2], C5×M5(2) [×4], C22×C40, C10×M5(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10 [×7], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], M5(2) [×2], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C2×M5(2), C2×C40 [×6], C22×C20, C5×M5(2) [×2], C22×C40, C10×M5(2)
Generators and relations
G = < a,b,c | a10=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
►On 160 points
Generators in S160
(1 50 117 142 91 47 71 18 157 107)(2 51 118 143 92 48 72 19 158 108)(3 52 119 144 93 33 73 20 159 109)(4 53 120 129 94 34 74 21 160 110)(5 54 121 130 95 35 75 22 145 111)(6 55 122 131 96 36 76 23 146 112)(7 56 123 132 81 37 77 24 147 97)(8 57 124 133 82 38 78 25 148 98)(9 58 125 134 83 39 79 26 149 99)(10 59 126 135 84 40 80 27 150 100)(11 60 127 136 85 41 65 28 151 101)(12 61 128 137 86 42 66 29 152 102)(13 62 113 138 87 43 67 30 153 103)(14 63 114 139 88 44 68 31 154 104)(15 64 115 140 89 45 69 32 155 105)(16 49 116 141 90 46 70 17 156 106)
(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 39)(2 48)(3 41)(4 34)(5 43)(6 36)(7 45)(8 38)(9 47)(10 40)(11 33)(12 42)(13 35)(14 44)(15 37)(16 46)(17 116)(18 125)(19 118)(20 127)(21 120)(22 113)(23 122)(24 115)(25 124)(26 117)(27 126)(28 119)(29 128)(30 121)(31 114)(32 123)(49 70)(50 79)(51 72)(52 65)(53 74)(54 67)(55 76)(56 69)(57 78)(58 71)(59 80)(60 73)(61 66)(62 75)(63 68)(64 77)(81 105)(82 98)(83 107)(84 100)(85 109)(86 102)(87 111)(88 104)(89 97)(90 106)(91 99)(92 108)(93 101)(94 110)(95 103)(96 112)(129 160)(130 153)(131 146)(132 155)(133 148)(134 157)(135 150)(136 159)(137 152)(138 145)(139 154)(140 147)(141 156)(142 149)(143 158)(144 151)
G:=sub<Sym(160)| (1,50,117,142,91,47,71,18,157,107)(2,51,118,143,92,48,72,19,158,108)(3,52,119,144,93,33,73,20,159,109)(4,53,120,129,94,34,74,21,160,110)(5,54,121,130,95,35,75,22,145,111)(6,55,122,131,96,36,76,23,146,112)(7,56,123,132,81,37,77,24,147,97)(8,57,124,133,82,38,78,25,148,98)(9,58,125,134,83,39,79,26,149,99)(10,59,126,135,84,40,80,27,150,100)(11,60,127,136,85,41,65,28,151,101)(12,61,128,137,86,42,66,29,152,102)(13,62,113,138,87,43,67,30,153,103)(14,63,114,139,88,44,68,31,154,104)(15,64,115,140,89,45,69,32,155,105)(16,49,116,141,90,46,70,17,156,106), (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), (1,39)(2,48)(3,41)(4,34)(5,43)(6,36)(7,45)(8,38)(9,47)(10,40)(11,33)(12,42)(13,35)(14,44)(15,37)(16,46)(17,116)(18,125)(19,118)(20,127)(21,120)(22,113)(23,122)(24,115)(25,124)(26,117)(27,126)(28,119)(29,128)(30,121)(31,114)(32,123)(49,70)(50,79)(51,72)(52,65)(53,74)(54,67)(55,76)(56,69)(57,78)(58,71)(59,80)(60,73)(61,66)(62,75)(63,68)(64,77)(81,105)(82,98)(83,107)(84,100)(85,109)(86,102)(87,111)(88,104)(89,97)(90,106)(91,99)(92,108)(93,101)(94,110)(95,103)(96,112)(129,160)(130,153)(131,146)(132,155)(133,148)(134,157)(135,150)(136,159)(137,152)(138,145)(139,154)(140,147)(141,156)(142,149)(143,158)(144,151)>;
G:=Group( (1,50,117,142,91,47,71,18,157,107)(2,51,118,143,92,48,72,19,158,108)(3,52,119,144,93,33,73,20,159,109)(4,53,120,129,94,34,74,21,160,110)(5,54,121,130,95,35,75,22,145,111)(6,55,122,131,96,36,76,23,146,112)(7,56,123,132,81,37,77,24,147,97)(8,57,124,133,82,38,78,25,148,98)(9,58,125,134,83,39,79,26,149,99)(10,59,126,135,84,40,80,27,150,100)(11,60,127,136,85,41,65,28,151,101)(12,61,128,137,86,42,66,29,152,102)(13,62,113,138,87,43,67,30,153,103)(14,63,114,139,88,44,68,31,154,104)(15,64,115,140,89,45,69,32,155,105)(16,49,116,141,90,46,70,17,156,106), (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), (1,39)(2,48)(3,41)(4,34)(5,43)(6,36)(7,45)(8,38)(9,47)(10,40)(11,33)(12,42)(13,35)(14,44)(15,37)(16,46)(17,116)(18,125)(19,118)(20,127)(21,120)(22,113)(23,122)(24,115)(25,124)(26,117)(27,126)(28,119)(29,128)(30,121)(31,114)(32,123)(49,70)(50,79)(51,72)(52,65)(53,74)(54,67)(55,76)(56,69)(57,78)(58,71)(59,80)(60,73)(61,66)(62,75)(63,68)(64,77)(81,105)(82,98)(83,107)(84,100)(85,109)(86,102)(87,111)(88,104)(89,97)(90,106)(91,99)(92,108)(93,101)(94,110)(95,103)(96,112)(129,160)(130,153)(131,146)(132,155)(133,148)(134,157)(135,150)(136,159)(137,152)(138,145)(139,154)(140,147)(141,156)(142,149)(143,158)(144,151) );
G=PermutationGroup([(1,50,117,142,91,47,71,18,157,107),(2,51,118,143,92,48,72,19,158,108),(3,52,119,144,93,33,73,20,159,109),(4,53,120,129,94,34,74,21,160,110),(5,54,121,130,95,35,75,22,145,111),(6,55,122,131,96,36,76,23,146,112),(7,56,123,132,81,37,77,24,147,97),(8,57,124,133,82,38,78,25,148,98),(9,58,125,134,83,39,79,26,149,99),(10,59,126,135,84,40,80,27,150,100),(11,60,127,136,85,41,65,28,151,101),(12,61,128,137,86,42,66,29,152,102),(13,62,113,138,87,43,67,30,153,103),(14,63,114,139,88,44,68,31,154,104),(15,64,115,140,89,45,69,32,155,105),(16,49,116,141,90,46,70,17,156,106)], [(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)], [(1,39),(2,48),(3,41),(4,34),(5,43),(6,36),(7,45),(8,38),(9,47),(10,40),(11,33),(12,42),(13,35),(14,44),(15,37),(16,46),(17,116),(18,125),(19,118),(20,127),(21,120),(22,113),(23,122),(24,115),(25,124),(26,117),(27,126),(28,119),(29,128),(30,121),(31,114),(32,123),(49,70),(50,79),(51,72),(52,65),(53,74),(54,67),(55,76),(56,69),(57,78),(58,71),(59,80),(60,73),(61,66),(62,75),(63,68),(64,77),(81,105),(82,98),(83,107),(84,100),(85,109),(86,102),(87,111),(88,104),(89,97),(90,106),(91,99),(92,108),(93,101),(94,110),(95,103),(96,112),(129,160),(130,153),(131,146),(132,155),(133,148),(134,157),(135,150),(136,159),(137,152),(138,145),(139,154),(140,147),(141,156),(142,149),(143,158),(144,151)])
Matrix representation ►G ⊆ GL3(𝔽241) generated by
| 240 | 0 | 0 |
| 0 | 150 | 0 |
| 0 | 0 | 150 |
| 240 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 8 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 240 |
G:=sub<GL(3,GF(241))| [240,0,0,0,150,0,0,0,150],[240,0,0,0,0,8,0,1,0],[1,0,0,0,1,0,0,0,240] >;
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | ··· | 10T | 16A | ··· | 16P | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40AF | 40AG | ··· | 40AV | 80A | ··· | 80BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | C40 | M5(2) | C5×M5(2) |
| kernel | C10×M5(2) | C2×C80 | C5×M5(2) | C22×C40 | C2×C40 | C22×C20 | C2×M5(2) | C2×C20 | C22×C10 | C2×C16 | M5(2) | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 4 | 12 | 4 | 8 | 16 | 4 | 24 | 8 | 48 | 16 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times M_{5(2)} % in TeX
G:=Group("C10xM5(2)"); // GroupNames label
G:=SmallGroup(320,1004);
// by ID
G=gap.SmallGroup(320,1004);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,2269,102,124]);
// 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^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀