metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.216D10, C4⋊C4.77D10, C42.C2⋊3D5, (C2×C20).276D4, C20.72(C4○D4), C10.Q16⋊41C2, C4.D20.9C2, D20⋊6C4.13C2, C10.110(C4○D8), (C2×C20).386C23, (C4×C20).116C22, C4.14(Q8⋊2D5), C10.56(C4.4D4), C2.9(C20.23D4), (C2×D20).108C22, C2.29(D4.8D10), C5⋊5(C42.78C22), (C2×Dic10).113C22, (C4×C5⋊2C8)⋊13C2, (C5×C42.C2)⋊3C2, (C2×C10).517(C2×D4), (C2×C4).112(C5⋊D4), (C5×C4⋊C4).124C22, (C2×C4).484(C22×D5), C22.190(C2×C5⋊D4), (C2×C5⋊2C8).263C22, SmallGroup(320,695)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.216D10
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c9 >
Subgroups: 398 in 96 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C5⋊2C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×D5, C42.78C22, C2×C5⋊2C8, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, C4×C5⋊2C8, D20⋊6C4, C10.Q16, C4.D20, C5×C42.C2, C42.216D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C4○D8, C5⋊D4, C22×D5, C42.78C22, Q8⋊2D5, C2×C5⋊D4, C20.23D4, D4.8D10, C42.216D10
►On 160 points
Generators in S160
(1 102 157 82)(2 113 158 93)(3 104 159 84)(4 115 160 95)(5 106 141 86)(6 117 142 97)(7 108 143 88)(8 119 144 99)(9 110 145 90)(10 101 146 81)(11 112 147 92)(12 103 148 83)(13 114 149 94)(14 105 150 85)(15 116 151 96)(16 107 152 87)(17 118 153 98)(18 109 154 89)(19 120 155 100)(20 111 156 91)(21 59 79 126)(22 50 80 137)(23 41 61 128)(24 52 62 139)(25 43 63 130)(26 54 64 121)(27 45 65 132)(28 56 66 123)(29 47 67 134)(30 58 68 125)(31 49 69 136)(32 60 70 127)(33 51 71 138)(34 42 72 129)(35 53 73 140)(36 44 74 131)(37 55 75 122)(38 46 76 133)(39 57 77 124)(40 48 78 135)
(1 48 147 125)(2 126 148 49)(3 50 149 127)(4 128 150 51)(5 52 151 129)(6 130 152 53)(7 54 153 131)(8 132 154 55)(9 56 155 133)(10 134 156 57)(11 58 157 135)(12 136 158 59)(13 60 159 137)(14 138 160 41)(15 42 141 139)(16 140 142 43)(17 44 143 121)(18 122 144 45)(19 46 145 123)(20 124 146 47)(21 83 69 113)(22 114 70 84)(23 85 71 115)(24 116 72 86)(25 87 73 117)(26 118 74 88)(27 89 75 119)(28 120 76 90)(29 91 77 101)(30 102 78 92)(31 93 79 103)(32 104 80 94)(33 95 61 105)(34 106 62 96)(35 97 63 107)(36 108 64 98)(37 99 65 109)(38 110 66 100)(39 81 67 111)(40 112 68 82)
(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 10 135 47 147 156 58 124)(2 123 59 155 148 46 136 9)(3 8 137 45 149 154 60 122)(4 121 41 153 150 44 138 7)(5 6 139 43 151 152 42 140)(11 20 125 57 157 146 48 134)(12 133 49 145 158 56 126 19)(13 18 127 55 159 144 50 132)(14 131 51 143 160 54 128 17)(15 16 129 53 141 142 52 130)(21 120 103 38 69 90 93 66)(22 65 94 89 70 37 104 119)(23 118 105 36 71 88 95 64)(24 63 96 87 72 35 106 117)(25 116 107 34 73 86 97 62)(26 61 98 85 74 33 108 115)(27 114 109 32 75 84 99 80)(28 79 100 83 76 31 110 113)(29 112 111 30 77 82 81 78)(39 102 101 40 67 92 91 68)
G:=sub<Sym(160)| (1,102,157,82)(2,113,158,93)(3,104,159,84)(4,115,160,95)(5,106,141,86)(6,117,142,97)(7,108,143,88)(8,119,144,99)(9,110,145,90)(10,101,146,81)(11,112,147,92)(12,103,148,83)(13,114,149,94)(14,105,150,85)(15,116,151,96)(16,107,152,87)(17,118,153,98)(18,109,154,89)(19,120,155,100)(20,111,156,91)(21,59,79,126)(22,50,80,137)(23,41,61,128)(24,52,62,139)(25,43,63,130)(26,54,64,121)(27,45,65,132)(28,56,66,123)(29,47,67,134)(30,58,68,125)(31,49,69,136)(32,60,70,127)(33,51,71,138)(34,42,72,129)(35,53,73,140)(36,44,74,131)(37,55,75,122)(38,46,76,133)(39,57,77,124)(40,48,78,135), (1,48,147,125)(2,126,148,49)(3,50,149,127)(4,128,150,51)(5,52,151,129)(6,130,152,53)(7,54,153,131)(8,132,154,55)(9,56,155,133)(10,134,156,57)(11,58,157,135)(12,136,158,59)(13,60,159,137)(14,138,160,41)(15,42,141,139)(16,140,142,43)(17,44,143,121)(18,122,144,45)(19,46,145,123)(20,124,146,47)(21,83,69,113)(22,114,70,84)(23,85,71,115)(24,116,72,86)(25,87,73,117)(26,118,74,88)(27,89,75,119)(28,120,76,90)(29,91,77,101)(30,102,78,92)(31,93,79,103)(32,104,80,94)(33,95,61,105)(34,106,62,96)(35,97,63,107)(36,108,64,98)(37,99,65,109)(38,110,66,100)(39,81,67,111)(40,112,68,82), (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), (1,10,135,47,147,156,58,124)(2,123,59,155,148,46,136,9)(3,8,137,45,149,154,60,122)(4,121,41,153,150,44,138,7)(5,6,139,43,151,152,42,140)(11,20,125,57,157,146,48,134)(12,133,49,145,158,56,126,19)(13,18,127,55,159,144,50,132)(14,131,51,143,160,54,128,17)(15,16,129,53,141,142,52,130)(21,120,103,38,69,90,93,66)(22,65,94,89,70,37,104,119)(23,118,105,36,71,88,95,64)(24,63,96,87,72,35,106,117)(25,116,107,34,73,86,97,62)(26,61,98,85,74,33,108,115)(27,114,109,32,75,84,99,80)(28,79,100,83,76,31,110,113)(29,112,111,30,77,82,81,78)(39,102,101,40,67,92,91,68)>;
G:=Group( (1,102,157,82)(2,113,158,93)(3,104,159,84)(4,115,160,95)(5,106,141,86)(6,117,142,97)(7,108,143,88)(8,119,144,99)(9,110,145,90)(10,101,146,81)(11,112,147,92)(12,103,148,83)(13,114,149,94)(14,105,150,85)(15,116,151,96)(16,107,152,87)(17,118,153,98)(18,109,154,89)(19,120,155,100)(20,111,156,91)(21,59,79,126)(22,50,80,137)(23,41,61,128)(24,52,62,139)(25,43,63,130)(26,54,64,121)(27,45,65,132)(28,56,66,123)(29,47,67,134)(30,58,68,125)(31,49,69,136)(32,60,70,127)(33,51,71,138)(34,42,72,129)(35,53,73,140)(36,44,74,131)(37,55,75,122)(38,46,76,133)(39,57,77,124)(40,48,78,135), (1,48,147,125)(2,126,148,49)(3,50,149,127)(4,128,150,51)(5,52,151,129)(6,130,152,53)(7,54,153,131)(8,132,154,55)(9,56,155,133)(10,134,156,57)(11,58,157,135)(12,136,158,59)(13,60,159,137)(14,138,160,41)(15,42,141,139)(16,140,142,43)(17,44,143,121)(18,122,144,45)(19,46,145,123)(20,124,146,47)(21,83,69,113)(22,114,70,84)(23,85,71,115)(24,116,72,86)(25,87,73,117)(26,118,74,88)(27,89,75,119)(28,120,76,90)(29,91,77,101)(30,102,78,92)(31,93,79,103)(32,104,80,94)(33,95,61,105)(34,106,62,96)(35,97,63,107)(36,108,64,98)(37,99,65,109)(38,110,66,100)(39,81,67,111)(40,112,68,82), (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), (1,10,135,47,147,156,58,124)(2,123,59,155,148,46,136,9)(3,8,137,45,149,154,60,122)(4,121,41,153,150,44,138,7)(5,6,139,43,151,152,42,140)(11,20,125,57,157,146,48,134)(12,133,49,145,158,56,126,19)(13,18,127,55,159,144,50,132)(14,131,51,143,160,54,128,17)(15,16,129,53,141,142,52,130)(21,120,103,38,69,90,93,66)(22,65,94,89,70,37,104,119)(23,118,105,36,71,88,95,64)(24,63,96,87,72,35,106,117)(25,116,107,34,73,86,97,62)(26,61,98,85,74,33,108,115)(27,114,109,32,75,84,99,80)(28,79,100,83,76,31,110,113)(29,112,111,30,77,82,81,78)(39,102,101,40,67,92,91,68) );
G=PermutationGroup([[(1,102,157,82),(2,113,158,93),(3,104,159,84),(4,115,160,95),(5,106,141,86),(6,117,142,97),(7,108,143,88),(8,119,144,99),(9,110,145,90),(10,101,146,81),(11,112,147,92),(12,103,148,83),(13,114,149,94),(14,105,150,85),(15,116,151,96),(16,107,152,87),(17,118,153,98),(18,109,154,89),(19,120,155,100),(20,111,156,91),(21,59,79,126),(22,50,80,137),(23,41,61,128),(24,52,62,139),(25,43,63,130),(26,54,64,121),(27,45,65,132),(28,56,66,123),(29,47,67,134),(30,58,68,125),(31,49,69,136),(32,60,70,127),(33,51,71,138),(34,42,72,129),(35,53,73,140),(36,44,74,131),(37,55,75,122),(38,46,76,133),(39,57,77,124),(40,48,78,135)], [(1,48,147,125),(2,126,148,49),(3,50,149,127),(4,128,150,51),(5,52,151,129),(6,130,152,53),(7,54,153,131),(8,132,154,55),(9,56,155,133),(10,134,156,57),(11,58,157,135),(12,136,158,59),(13,60,159,137),(14,138,160,41),(15,42,141,139),(16,140,142,43),(17,44,143,121),(18,122,144,45),(19,46,145,123),(20,124,146,47),(21,83,69,113),(22,114,70,84),(23,85,71,115),(24,116,72,86),(25,87,73,117),(26,118,74,88),(27,89,75,119),(28,120,76,90),(29,91,77,101),(30,102,78,92),(31,93,79,103),(32,104,80,94),(33,95,61,105),(34,106,62,96),(35,97,63,107),(36,108,64,98),(37,99,65,109),(38,110,66,100),(39,81,67,111),(40,112,68,82)], [(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)], [(1,10,135,47,147,156,58,124),(2,123,59,155,148,46,136,9),(3,8,137,45,149,154,60,122),(4,121,41,153,150,44,138,7),(5,6,139,43,151,152,42,140),(11,20,125,57,157,146,48,134),(12,133,49,145,158,56,126,19),(13,18,127,55,159,144,50,132),(14,131,51,143,160,54,128,17),(15,16,129,53,141,142,52,130),(21,120,103,38,69,90,93,66),(22,65,94,89,70,37,104,119),(23,118,105,36,71,88,95,64),(24,63,96,87,72,35,106,117),(25,116,107,34,73,86,97,62),(26,61,98,85,74,33,108,115),(27,114,109,32,75,84,99,80),(28,79,100,83,76,31,110,113),(29,112,111,30,77,82,81,78),(39,102,101,40,67,92,91,68)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | ··· | 2 | 8 | 8 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C5⋊D4 | Q8⋊2D5 | D4.8D10 |
| kernel | C42.216D10 | C4×C5⋊2C8 | D20⋊6C4 | C10.Q16 | C4.D20 | C5×C42.C2 | C2×C20 | C42.C2 | C20 | C42 | C4⋊C4 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C42.216D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 25 |
| 0 | 0 | 0 | 0 | 5 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 21 |
| 0 | 0 | 0 | 0 | 37 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 38 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 19 | 0 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 3 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 19 | 0 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,5,0,0,0,0,25,9],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,37,0,0,0,0,21,1],[12,12,0,0,0,0,12,29,0,0,0,0,0,0,24,3,0,0,0,0,38,3,0,0,0,0,0,0,0,19,0,0,0,0,28,0],[12,12,0,0,0,0,29,12,0,0,0,0,0,0,24,3,0,0,0,0,40,17,0,0,0,0,0,0,30,19,0,0,0,0,13,0] >;
C42.216D10 in GAP, Magma, Sage, TeX
C_4^2._{216}D_{10} % in TeX
G:=Group("C4^2.216D10"); // GroupNames label
G:=SmallGroup(320,695);
// by ID
G=gap.SmallGroup(320,695);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^9>;
// generators/relations