metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.200D10, Dic5.15M4(2), C4⋊C8⋊17D5, (C4×D5)⋊2C8, C4.14(C8×D5), C20.35(C2×C8), D10.9(C2×C8), C20⋊3C8⋊12C2, (C8×Dic5)⋊23C2, (C2×C8).215D10, (C4×Dic5).9C4, (D5×C42).2C2, C2.6(D5×M4(2)), D10⋊1C8.7C2, (C4×C20).59C22, C10.34(C22×C8), Dic5.21(C2×C8), C20.304(C4○D4), (C2×C40).209C22, (C2×C20).830C23, C5⋊7(C42.12C4), C4.52(Q8⋊2D5), C10.60(C2×M4(2)), C4.130(D4⋊2D5), C10.50(C42⋊C2), (C4×Dic5).359C22, (C5×C4⋊C8)⋊14C2, C2.12(D5×C2×C8), (C2×C4×D5).10C4, C22.47(C2×C4×D5), (C2×C4).145(C4×D5), (C2×C20).242(C2×C4), C2.3(C4⋊C4⋊7D5), (C2×C4×D5).346C22, (C2×C4).772(C22×D5), (C2×C10).186(C22×C4), (C2×C5⋊2C8).313C22, (C2×Dic5).143(C2×C4), (C22×D5).101(C2×C4), SmallGroup(320,460)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.200D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2c9 >
Subgroups: 350 in 118 conjugacy classes, 61 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×2], C2×C8 [×2], C22×C4 [×3], Dic5 [×4], Dic5, C20 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C22⋊C8 [×2], C4⋊C8, C4⋊C8, C2×C42, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.12C4, C2×C5⋊2C8 [×2], C4×Dic5 [×3], C4×C20, C2×C40 [×2], C2×C4×D5 [×3], C20⋊3C8, C8×Dic5 [×2], D10⋊1C8 [×2], C5×C4⋊C8, D5×C42, C42.200D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C22×C8, C2×M4(2), C4×D5 [×2], C22×D5, C42.12C4, C8×D5 [×2], C2×C4×D5, D4⋊2D5, Q8⋊2D5, C4⋊C4⋊7D5, D5×C2×C8, D5×M4(2), C42.200D10
►On 160 points
Generators in S160
(1 113 71 153)(2 154 72 114)(3 115 73 155)(4 156 74 116)(5 117 75 157)(6 158 76 118)(7 119 77 159)(8 160 78 120)(9 81 79 121)(10 122 80 82)(11 83 41 123)(12 124 42 84)(13 85 43 125)(14 126 44 86)(15 87 45 127)(16 128 46 88)(17 89 47 129)(18 130 48 90)(19 91 49 131)(20 132 50 92)(21 93 51 133)(22 134 52 94)(23 95 53 135)(24 136 54 96)(25 97 55 137)(26 138 56 98)(27 99 57 139)(28 140 58 100)(29 101 59 141)(30 142 60 102)(31 103 61 143)(32 144 62 104)(33 105 63 145)(34 146 64 106)(35 107 65 147)(36 148 66 108)(37 109 67 149)(38 150 68 110)(39 111 69 151)(40 152 70 112)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 10 41 50 21 30 61 70)(2 49 42 29 22 69 62 9)(3 28 43 68 23 8 63 48)(4 67 44 7 24 47 64 27)(5 6 45 46 25 26 65 66)(11 20 51 60 31 40 71 80)(12 59 52 39 32 79 72 19)(13 38 53 78 33 18 73 58)(14 77 54 17 34 57 74 37)(15 16 55 56 35 36 75 76)(81 114 131 124 101 94 151 144)(82 123 132 93 102 143 152 113)(83 92 133 142 103 112 153 122)(84 141 134 111 104 121 154 91)(85 110 135 160 105 90 155 140)(86 159 136 89 106 139 156 109)(87 88 137 138 107 108 157 158)(95 120 145 130 115 100 125 150)(96 129 146 99 116 149 126 119)(97 98 147 148 117 118 127 128)
G:=sub<Sym(160)| (1,113,71,153)(2,154,72,114)(3,115,73,155)(4,156,74,116)(5,117,75,157)(6,158,76,118)(7,119,77,159)(8,160,78,120)(9,81,79,121)(10,122,80,82)(11,83,41,123)(12,124,42,84)(13,85,43,125)(14,126,44,86)(15,87,45,127)(16,128,46,88)(17,89,47,129)(18,130,48,90)(19,91,49,131)(20,132,50,92)(21,93,51,133)(22,134,52,94)(23,95,53,135)(24,136,54,96)(25,97,55,137)(26,138,56,98)(27,99,57,139)(28,140,58,100)(29,101,59,141)(30,142,60,102)(31,103,61,143)(32,144,62,104)(33,105,63,145)(34,146,64,106)(35,107,65,147)(36,148,66,108)(37,109,67,149)(38,150,68,110)(39,111,69,151)(40,152,70,112), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,41,50,21,30,61,70)(2,49,42,29,22,69,62,9)(3,28,43,68,23,8,63,48)(4,67,44,7,24,47,64,27)(5,6,45,46,25,26,65,66)(11,20,51,60,31,40,71,80)(12,59,52,39,32,79,72,19)(13,38,53,78,33,18,73,58)(14,77,54,17,34,57,74,37)(15,16,55,56,35,36,75,76)(81,114,131,124,101,94,151,144)(82,123,132,93,102,143,152,113)(83,92,133,142,103,112,153,122)(84,141,134,111,104,121,154,91)(85,110,135,160,105,90,155,140)(86,159,136,89,106,139,156,109)(87,88,137,138,107,108,157,158)(95,120,145,130,115,100,125,150)(96,129,146,99,116,149,126,119)(97,98,147,148,117,118,127,128)>;
G:=Group( (1,113,71,153)(2,154,72,114)(3,115,73,155)(4,156,74,116)(5,117,75,157)(6,158,76,118)(7,119,77,159)(8,160,78,120)(9,81,79,121)(10,122,80,82)(11,83,41,123)(12,124,42,84)(13,85,43,125)(14,126,44,86)(15,87,45,127)(16,128,46,88)(17,89,47,129)(18,130,48,90)(19,91,49,131)(20,132,50,92)(21,93,51,133)(22,134,52,94)(23,95,53,135)(24,136,54,96)(25,97,55,137)(26,138,56,98)(27,99,57,139)(28,140,58,100)(29,101,59,141)(30,142,60,102)(31,103,61,143)(32,144,62,104)(33,105,63,145)(34,146,64,106)(35,107,65,147)(36,148,66,108)(37,109,67,149)(38,150,68,110)(39,111,69,151)(40,152,70,112), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,41,50,21,30,61,70)(2,49,42,29,22,69,62,9)(3,28,43,68,23,8,63,48)(4,67,44,7,24,47,64,27)(5,6,45,46,25,26,65,66)(11,20,51,60,31,40,71,80)(12,59,52,39,32,79,72,19)(13,38,53,78,33,18,73,58)(14,77,54,17,34,57,74,37)(15,16,55,56,35,36,75,76)(81,114,131,124,101,94,151,144)(82,123,132,93,102,143,152,113)(83,92,133,142,103,112,153,122)(84,141,134,111,104,121,154,91)(85,110,135,160,105,90,155,140)(86,159,136,89,106,139,156,109)(87,88,137,138,107,108,157,158)(95,120,145,130,115,100,125,150)(96,129,146,99,116,149,126,119)(97,98,147,148,117,118,127,128) );
G=PermutationGroup([(1,113,71,153),(2,154,72,114),(3,115,73,155),(4,156,74,116),(5,117,75,157),(6,158,76,118),(7,119,77,159),(8,160,78,120),(9,81,79,121),(10,122,80,82),(11,83,41,123),(12,124,42,84),(13,85,43,125),(14,126,44,86),(15,87,45,127),(16,128,46,88),(17,89,47,129),(18,130,48,90),(19,91,49,131),(20,132,50,92),(21,93,51,133),(22,134,52,94),(23,95,53,135),(24,136,54,96),(25,97,55,137),(26,138,56,98),(27,99,57,139),(28,140,58,100),(29,101,59,141),(30,142,60,102),(31,103,61,143),(32,144,62,104),(33,105,63,145),(34,146,64,106),(35,107,65,147),(36,148,66,108),(37,109,67,149),(38,150,68,110),(39,111,69,151),(40,152,70,112)], [(1,11,21,31),(2,12,22,32),(3,13,23,33),(4,14,24,34),(5,15,25,35),(6,16,26,36),(7,17,27,37),(8,18,28,38),(9,19,29,39),(10,20,30,40),(41,51,61,71),(42,52,62,72),(43,53,63,73),(44,54,64,74),(45,55,65,75),(46,56,66,76),(47,57,67,77),(48,58,68,78),(49,59,69,79),(50,60,70,80),(81,91,101,111),(82,92,102,112),(83,93,103,113),(84,94,104,114),(85,95,105,115),(86,96,106,116),(87,97,107,117),(88,98,108,118),(89,99,109,119),(90,100,110,120),(121,131,141,151),(122,132,142,152),(123,133,143,153),(124,134,144,154),(125,135,145,155),(126,136,146,156),(127,137,147,157),(128,138,148,158),(129,139,149,159),(130,140,150,160)], [(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)], [(1,10,41,50,21,30,61,70),(2,49,42,29,22,69,62,9),(3,28,43,68,23,8,63,48),(4,67,44,7,24,47,64,27),(5,6,45,46,25,26,65,66),(11,20,51,60,31,40,71,80),(12,59,52,39,32,79,72,19),(13,38,53,78,33,18,73,58),(14,77,54,17,34,57,74,37),(15,16,55,56,35,36,75,76),(81,114,131,124,101,94,151,144),(82,123,132,93,102,143,152,113),(83,92,133,142,103,112,153,122),(84,141,134,111,104,121,154,91),(85,110,135,160,105,90,155,140),(86,159,136,89,106,139,156,109),(87,88,137,138,107,108,157,158),(95,120,145,130,115,100,125,150),(96,129,146,99,116,149,126,119),(97,98,147,148,117,118,127,128)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D5 | M4(2) | C4○D4 | D10 | D10 | C4×D5 | C8×D5 | D4⋊2D5 | Q8⋊2D5 | D5×M4(2) |
| kernel | C42.200D10 | C20⋊3C8 | C8×Dic5 | D10⋊1C8 | C5×C4⋊C8 | D5×C42 | C4×Dic5 | C2×C4×D5 | C4×D5 | C4⋊C8 | Dic5 | C20 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 16 | 2 | 4 | 4 | 2 | 4 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of C42.200D10 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 6 | 6 | 0 | 0 |
| 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 14 | 0 | 0 | 0 | 0 |
| 0 | 6 | 6 | 0 | 0 |
| 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,32,0,0,0,0,0,9],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[27,0,0,0,0,0,6,35,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,1,0],[14,0,0,0,0,0,6,1,0,0,0,6,35,0,0,0,0,0,0,40,0,0,0,40,0] >;
C42.200D10 in GAP, Magma, Sage, TeX
C_4^2._{200}D_{10} % in TeX
G:=Group("C4^2.200D10"); // GroupNames label
G:=SmallGroup(320,460);
// by ID
G=gap.SmallGroup(320,460);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,422,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations