direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C8○D4, C20.93C24, C40.80C23, C4○D4.4C20, D4.8(C2×C20), Q8.9(C2×C20), (C2×C40)⋊54C22, (C22×C40)⋊27C2, (C22×C8)⋊13C10, (C2×D4).12C20, (D4×C10).37C4, (Q8×C10).30C4, (C2×Q8).10C20, C4.17(C23×C10), C2.11(C23×C20), C4.22(C22×C20), C10.84(C23×C4), C23.20(C2×C20), C8.17(C22×C10), M4(2)⋊11(C2×C10), (C10×M4(2))⋊35C2, (C2×M4(2))⋊17C10, (C2×C20).969C23, C20.226(C22×C4), C22.4(C22×C20), (C5×M4(2))⋊40C22, (C22×C20).600C22, (C2×C8)⋊16(C2×C10), (C2×C4).53(C2×C20), (C5×C4○D4).12C4, (C5×D4).44(C2×C4), (C5×Q8).48(C2×C4), (C2×C20).447(C2×C4), (C10×C4○D4).28C2, (C2×C4○D4).14C10, C4○D4.14(C2×C10), (C5×C4○D4).59C22, (C22×C10).154(C2×C4), (C22×C4).127(C2×C10), (C2×C10).136(C22×C4), (C2×C4).139(C22×C10), SmallGroup(320,1569)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 290 in 266 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C40 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C2×C8○D4, C2×C40, C2×C40 [×15], C5×M4(2) [×12], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C22×C40 [×3], C10×M4(2) [×3], C5×C8○D4 [×8], C10×C4○D4, C10×C8○D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C8○D4 [×2], C23×C4, C2×C20 [×28], C22×C10 [×15], C2×C8○D4, C22×C20 [×14], C23×C10, C5×C8○D4 [×2], C23×C20, C10×C8○D4
Generators and relations
G = < a,b,c,d | a10=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
►On 160 points
Generators in S160
(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)
(1 125 67 148 59 134 50 119)(2 126 68 149 60 135 41 120)(3 127 69 150 51 136 42 111)(4 128 70 141 52 137 43 112)(5 129 61 142 53 138 44 113)(6 130 62 143 54 139 45 114)(7 121 63 144 55 140 46 115)(8 122 64 145 56 131 47 116)(9 123 65 146 57 132 48 117)(10 124 66 147 58 133 49 118)(11 74 37 90 159 103 28 99)(12 75 38 81 160 104 29 100)(13 76 39 82 151 105 30 91)(14 77 40 83 152 106 21 92)(15 78 31 84 153 107 22 93)(16 79 32 85 154 108 23 94)(17 80 33 86 155 109 24 95)(18 71 34 87 156 110 25 96)(19 72 35 88 157 101 26 97)(20 73 36 89 158 102 27 98)
(1 74 59 103)(2 75 60 104)(3 76 51 105)(4 77 52 106)(5 78 53 107)(6 79 54 108)(7 80 55 109)(8 71 56 110)(9 72 57 101)(10 73 58 102)(11 148 159 119)(12 149 160 120)(13 150 151 111)(14 141 152 112)(15 142 153 113)(16 143 154 114)(17 144 155 115)(18 145 156 116)(19 146 157 117)(20 147 158 118)(21 128 40 137)(22 129 31 138)(23 130 32 139)(24 121 33 140)(25 122 34 131)(26 123 35 132)(27 124 36 133)(28 125 37 134)(29 126 38 135)(30 127 39 136)(41 100 68 81)(42 91 69 82)(43 92 70 83)(44 93 61 84)(45 94 62 85)(46 95 63 86)(47 96 64 87)(48 97 65 88)(49 98 66 89)(50 99 67 90)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 111)(19 112)(20 113)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 138)(28 139)(29 140)(30 131)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 121)(39 122)(40 123)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 91)(48 92)(49 93)(50 94)(51 110)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(61 89)(62 90)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(141 157)(142 158)(143 159)(144 160)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)
G:=sub<Sym(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), (1,125,67,148,59,134,50,119)(2,126,68,149,60,135,41,120)(3,127,69,150,51,136,42,111)(4,128,70,141,52,137,43,112)(5,129,61,142,53,138,44,113)(6,130,62,143,54,139,45,114)(7,121,63,144,55,140,46,115)(8,122,64,145,56,131,47,116)(9,123,65,146,57,132,48,117)(10,124,66,147,58,133,49,118)(11,74,37,90,159,103,28,99)(12,75,38,81,160,104,29,100)(13,76,39,82,151,105,30,91)(14,77,40,83,152,106,21,92)(15,78,31,84,153,107,22,93)(16,79,32,85,154,108,23,94)(17,80,33,86,155,109,24,95)(18,71,34,87,156,110,25,96)(19,72,35,88,157,101,26,97)(20,73,36,89,158,102,27,98), (1,74,59,103)(2,75,60,104)(3,76,51,105)(4,77,52,106)(5,78,53,107)(6,79,54,108)(7,80,55,109)(8,71,56,110)(9,72,57,101)(10,73,58,102)(11,148,159,119)(12,149,160,120)(13,150,151,111)(14,141,152,112)(15,142,153,113)(16,143,154,114)(17,144,155,115)(18,145,156,116)(19,146,157,117)(20,147,158,118)(21,128,40,137)(22,129,31,138)(23,130,32,139)(24,121,33,140)(25,122,34,131)(26,123,35,132)(27,124,36,133)(28,125,37,134)(29,126,38,135)(30,127,39,136)(41,100,68,81)(42,91,69,82)(43,92,70,83)(44,93,61,84)(45,94,62,85)(46,95,63,86)(47,96,64,87)(48,97,65,88)(49,98,66,89)(50,99,67,90), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,111)(19,112)(20,113)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,131)(31,124)(32,125)(33,126)(34,127)(35,128)(36,129)(37,130)(38,121)(39,122)(40,123)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,91)(48,92)(49,93)(50,94)(51,110)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(141,157)(142,158)(143,159)(144,160)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)>;
G:=Group( (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), (1,125,67,148,59,134,50,119)(2,126,68,149,60,135,41,120)(3,127,69,150,51,136,42,111)(4,128,70,141,52,137,43,112)(5,129,61,142,53,138,44,113)(6,130,62,143,54,139,45,114)(7,121,63,144,55,140,46,115)(8,122,64,145,56,131,47,116)(9,123,65,146,57,132,48,117)(10,124,66,147,58,133,49,118)(11,74,37,90,159,103,28,99)(12,75,38,81,160,104,29,100)(13,76,39,82,151,105,30,91)(14,77,40,83,152,106,21,92)(15,78,31,84,153,107,22,93)(16,79,32,85,154,108,23,94)(17,80,33,86,155,109,24,95)(18,71,34,87,156,110,25,96)(19,72,35,88,157,101,26,97)(20,73,36,89,158,102,27,98), (1,74,59,103)(2,75,60,104)(3,76,51,105)(4,77,52,106)(5,78,53,107)(6,79,54,108)(7,80,55,109)(8,71,56,110)(9,72,57,101)(10,73,58,102)(11,148,159,119)(12,149,160,120)(13,150,151,111)(14,141,152,112)(15,142,153,113)(16,143,154,114)(17,144,155,115)(18,145,156,116)(19,146,157,117)(20,147,158,118)(21,128,40,137)(22,129,31,138)(23,130,32,139)(24,121,33,140)(25,122,34,131)(26,123,35,132)(27,124,36,133)(28,125,37,134)(29,126,38,135)(30,127,39,136)(41,100,68,81)(42,91,69,82)(43,92,70,83)(44,93,61,84)(45,94,62,85)(46,95,63,86)(47,96,64,87)(48,97,65,88)(49,98,66,89)(50,99,67,90), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,111)(19,112)(20,113)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,131)(31,124)(32,125)(33,126)(34,127)(35,128)(36,129)(37,130)(38,121)(39,122)(40,123)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,91)(48,92)(49,93)(50,94)(51,110)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(141,157)(142,158)(143,159)(144,160)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156) );
G=PermutationGroup([(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)], [(1,125,67,148,59,134,50,119),(2,126,68,149,60,135,41,120),(3,127,69,150,51,136,42,111),(4,128,70,141,52,137,43,112),(5,129,61,142,53,138,44,113),(6,130,62,143,54,139,45,114),(7,121,63,144,55,140,46,115),(8,122,64,145,56,131,47,116),(9,123,65,146,57,132,48,117),(10,124,66,147,58,133,49,118),(11,74,37,90,159,103,28,99),(12,75,38,81,160,104,29,100),(13,76,39,82,151,105,30,91),(14,77,40,83,152,106,21,92),(15,78,31,84,153,107,22,93),(16,79,32,85,154,108,23,94),(17,80,33,86,155,109,24,95),(18,71,34,87,156,110,25,96),(19,72,35,88,157,101,26,97),(20,73,36,89,158,102,27,98)], [(1,74,59,103),(2,75,60,104),(3,76,51,105),(4,77,52,106),(5,78,53,107),(6,79,54,108),(7,80,55,109),(8,71,56,110),(9,72,57,101),(10,73,58,102),(11,148,159,119),(12,149,160,120),(13,150,151,111),(14,141,152,112),(15,142,153,113),(16,143,154,114),(17,144,155,115),(18,145,156,116),(19,146,157,117),(20,147,158,118),(21,128,40,137),(22,129,31,138),(23,130,32,139),(24,121,33,140),(25,122,34,131),(26,123,35,132),(27,124,36,133),(28,125,37,134),(29,126,38,135),(30,127,39,136),(41,100,68,81),(42,91,69,82),(43,92,70,83),(44,93,61,84),(45,94,62,85),(46,95,63,86),(47,96,64,87),(48,97,65,88),(49,98,66,89),(50,99,67,90)], [(1,79),(2,80),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,111),(19,112),(20,113),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,138),(28,139),(29,140),(30,131),(31,124),(32,125),(33,126),(34,127),(35,128),(36,129),(37,130),(38,121),(39,122),(40,123),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,91),(48,92),(49,93),(50,94),(51,110),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,109),(61,89),(62,90),(63,81),(64,82),(65,83),(66,84),(67,85),(68,86),(69,87),(70,88),(141,157),(142,158),(143,159),(144,160),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156)])
Matrix representation ►G ⊆ GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 14 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 40 |
| 0 | 40 | 0 |
G:=sub<GL(3,GF(41))| [40,0,0,0,16,0,0,0,16],[1,0,0,0,14,0,0,0,14],[1,0,0,0,0,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | ··· | 8T | 10A | ··· | 10L | 10M | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20AN | 40A | ··· | 40AF | 40AG | ··· | 40CB |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 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 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | C8○D4 | C5×C8○D4 |
| kernel | C10×C8○D4 | C22×C40 | C10×M4(2) | C5×C8○D4 | C10×C4○D4 | D4×C10 | Q8×C10 | C5×C4○D4 | C2×C8○D4 | C22×C8 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C10 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 2 | 8 | 4 | 12 | 12 | 32 | 4 | 24 | 8 | 32 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times C_8\circ D_4 % in TeX
G:=Group("C10xC8oD4"); // GroupNames label
G:=SmallGroup(320,1569);
// by ID
G=gap.SmallGroup(320,1569);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1731,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀