metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.214D10, C4.12(D4×D5), C4.4D4⋊3D5, C20.26(C2×D4), C5⋊2C8.29D4, (C2×D4).49D10, (C2×C20).272D4, C5⋊2(C8.12D4), (C2×Q8).39D10, C4.D20⋊14C2, C2.9(C20⋊D4), C10.106(C4○D8), C10.18(C4⋊1D4), (C4×C20).108C22, (C2×C20).377C23, (D4×C10).65C22, (Q8×C10).57C22, (C2×D20).106C22, C2.25(D4.8D10), (C2×Dic10).111C22, (C4×C5⋊2C8)⋊12C2, (C2×Q8⋊D5)⋊14C2, (C2×D4⋊D5).7C2, (C2×D4.D5)⋊12C2, (C2×C5⋊Q16)⋊13C2, (C5×C4.4D4)⋊3C2, (C2×C10).508(C2×D4), (C2×C4).110(C5⋊D4), (C2×C4).477(C22×D5), C22.183(C2×C5⋊D4), (C2×C5⋊2C8).261C22, SmallGroup(320,686)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.214D10
G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, ad=da, bd=db, dcd-1=b-1c-1 >
Subgroups: 542 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, C4.4D4, C4.4D4, C2×D8, C2×SD16, C2×Q16, C5⋊2C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C8.12D4, C2×C5⋊2C8, D10⋊C4, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4×C20, C5×C22⋊C4, C2×Dic10, C2×D20, D4×C10, Q8×C10, C4×C5⋊2C8, C4.D20, C2×D4⋊D5, C2×D4.D5, C2×Q8⋊D5, C2×C5⋊Q16, C5×C4.4D4, C42.214D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C4○D8, C5⋊D4, C22×D5, C8.12D4, D4×D5, C2×C5⋊D4, C20⋊D4, D4.8D10, C42.214D10
►On 160 points
Generators in S160
(1 143 40 131)(2 101 31 119)(3 145 32 133)(4 103 33 111)(5 147 34 135)(6 105 35 113)(7 149 36 137)(8 107 37 115)(9 141 38 139)(10 109 39 117)(11 151 98 126)(12 45 99 89)(13 153 100 128)(14 47 91 81)(15 155 92 130)(16 49 93 83)(17 157 94 122)(18 41 95 85)(19 159 96 124)(20 43 97 87)(21 146 66 134)(22 104 67 112)(23 148 68 136)(24 106 69 114)(25 150 70 138)(26 108 61 116)(27 142 62 140)(28 110 63 118)(29 144 64 132)(30 102 65 120)(42 79 86 60)(44 71 88 52)(46 73 90 54)(48 75 82 56)(50 77 84 58)(51 160 80 125)(53 152 72 127)(55 154 74 129)(57 156 76 121)(59 158 78 123)
(1 73 63 13)(2 14 64 74)(3 75 65 15)(4 16 66 76)(5 77 67 17)(6 18 68 78)(7 79 69 19)(8 20 70 80)(9 71 61 11)(10 12 62 72)(21 57 33 93)(22 94 34 58)(23 59 35 95)(24 96 36 60)(25 51 37 97)(26 98 38 52)(27 53 39 99)(28 100 40 54)(29 55 31 91)(30 92 32 56)(41 136 123 105)(42 106 124 137)(43 138 125 107)(44 108 126 139)(45 140 127 109)(46 110 128 131)(47 132 129 101)(48 102 130 133)(49 134 121 103)(50 104 122 135)(81 144 154 119)(82 120 155 145)(83 146 156 111)(84 112 157 147)(85 148 158 113)(86 114 159 149)(87 150 160 115)(88 116 151 141)(89 142 152 117)(90 118 153 143)
(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 13 72 63 62 73 12)(2 11 74 61 64 71 14 9)(3 8 15 80 65 70 75 20)(4 19 76 69 66 79 16 7)(5 6 17 78 67 68 77 18)(21 60 93 36 33 96 57 24)(22 23 58 95 34 35 94 59)(25 56 97 32 37 92 51 30)(26 29 52 91 38 31 98 55)(27 54 99 40 39 100 53 28)(41 147 105 157 123 112 136 84)(42 83 137 111 124 156 106 146)(43 145 107 155 125 120 138 82)(44 81 139 119 126 154 108 144)(45 143 109 153 127 118 140 90)(46 89 131 117 128 152 110 142)(47 141 101 151 129 116 132 88)(48 87 133 115 130 160 102 150)(49 149 103 159 121 114 134 86)(50 85 135 113 122 158 104 148)
G:=sub<Sym(160)| (1,143,40,131)(2,101,31,119)(3,145,32,133)(4,103,33,111)(5,147,34,135)(6,105,35,113)(7,149,36,137)(8,107,37,115)(9,141,38,139)(10,109,39,117)(11,151,98,126)(12,45,99,89)(13,153,100,128)(14,47,91,81)(15,155,92,130)(16,49,93,83)(17,157,94,122)(18,41,95,85)(19,159,96,124)(20,43,97,87)(21,146,66,134)(22,104,67,112)(23,148,68,136)(24,106,69,114)(25,150,70,138)(26,108,61,116)(27,142,62,140)(28,110,63,118)(29,144,64,132)(30,102,65,120)(42,79,86,60)(44,71,88,52)(46,73,90,54)(48,75,82,56)(50,77,84,58)(51,160,80,125)(53,152,72,127)(55,154,74,129)(57,156,76,121)(59,158,78,123), (1,73,63,13)(2,14,64,74)(3,75,65,15)(4,16,66,76)(5,77,67,17)(6,18,68,78)(7,79,69,19)(8,20,70,80)(9,71,61,11)(10,12,62,72)(21,57,33,93)(22,94,34,58)(23,59,35,95)(24,96,36,60)(25,51,37,97)(26,98,38,52)(27,53,39,99)(28,100,40,54)(29,55,31,91)(30,92,32,56)(41,136,123,105)(42,106,124,137)(43,138,125,107)(44,108,126,139)(45,140,127,109)(46,110,128,131)(47,132,129,101)(48,102,130,133)(49,134,121,103)(50,104,122,135)(81,144,154,119)(82,120,155,145)(83,146,156,111)(84,112,157,147)(85,148,158,113)(86,114,159,149)(87,150,160,115)(88,116,151,141)(89,142,152,117)(90,118,153,143), (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), (1,10,13,72,63,62,73,12)(2,11,74,61,64,71,14,9)(3,8,15,80,65,70,75,20)(4,19,76,69,66,79,16,7)(5,6,17,78,67,68,77,18)(21,60,93,36,33,96,57,24)(22,23,58,95,34,35,94,59)(25,56,97,32,37,92,51,30)(26,29,52,91,38,31,98,55)(27,54,99,40,39,100,53,28)(41,147,105,157,123,112,136,84)(42,83,137,111,124,156,106,146)(43,145,107,155,125,120,138,82)(44,81,139,119,126,154,108,144)(45,143,109,153,127,118,140,90)(46,89,131,117,128,152,110,142)(47,141,101,151,129,116,132,88)(48,87,133,115,130,160,102,150)(49,149,103,159,121,114,134,86)(50,85,135,113,122,158,104,148)>;
G:=Group( (1,143,40,131)(2,101,31,119)(3,145,32,133)(4,103,33,111)(5,147,34,135)(6,105,35,113)(7,149,36,137)(8,107,37,115)(9,141,38,139)(10,109,39,117)(11,151,98,126)(12,45,99,89)(13,153,100,128)(14,47,91,81)(15,155,92,130)(16,49,93,83)(17,157,94,122)(18,41,95,85)(19,159,96,124)(20,43,97,87)(21,146,66,134)(22,104,67,112)(23,148,68,136)(24,106,69,114)(25,150,70,138)(26,108,61,116)(27,142,62,140)(28,110,63,118)(29,144,64,132)(30,102,65,120)(42,79,86,60)(44,71,88,52)(46,73,90,54)(48,75,82,56)(50,77,84,58)(51,160,80,125)(53,152,72,127)(55,154,74,129)(57,156,76,121)(59,158,78,123), (1,73,63,13)(2,14,64,74)(3,75,65,15)(4,16,66,76)(5,77,67,17)(6,18,68,78)(7,79,69,19)(8,20,70,80)(9,71,61,11)(10,12,62,72)(21,57,33,93)(22,94,34,58)(23,59,35,95)(24,96,36,60)(25,51,37,97)(26,98,38,52)(27,53,39,99)(28,100,40,54)(29,55,31,91)(30,92,32,56)(41,136,123,105)(42,106,124,137)(43,138,125,107)(44,108,126,139)(45,140,127,109)(46,110,128,131)(47,132,129,101)(48,102,130,133)(49,134,121,103)(50,104,122,135)(81,144,154,119)(82,120,155,145)(83,146,156,111)(84,112,157,147)(85,148,158,113)(86,114,159,149)(87,150,160,115)(88,116,151,141)(89,142,152,117)(90,118,153,143), (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), (1,10,13,72,63,62,73,12)(2,11,74,61,64,71,14,9)(3,8,15,80,65,70,75,20)(4,19,76,69,66,79,16,7)(5,6,17,78,67,68,77,18)(21,60,93,36,33,96,57,24)(22,23,58,95,34,35,94,59)(25,56,97,32,37,92,51,30)(26,29,52,91,38,31,98,55)(27,54,99,40,39,100,53,28)(41,147,105,157,123,112,136,84)(42,83,137,111,124,156,106,146)(43,145,107,155,125,120,138,82)(44,81,139,119,126,154,108,144)(45,143,109,153,127,118,140,90)(46,89,131,117,128,152,110,142)(47,141,101,151,129,116,132,88)(48,87,133,115,130,160,102,150)(49,149,103,159,121,114,134,86)(50,85,135,113,122,158,104,148) );
G=PermutationGroup([[(1,143,40,131),(2,101,31,119),(3,145,32,133),(4,103,33,111),(5,147,34,135),(6,105,35,113),(7,149,36,137),(8,107,37,115),(9,141,38,139),(10,109,39,117),(11,151,98,126),(12,45,99,89),(13,153,100,128),(14,47,91,81),(15,155,92,130),(16,49,93,83),(17,157,94,122),(18,41,95,85),(19,159,96,124),(20,43,97,87),(21,146,66,134),(22,104,67,112),(23,148,68,136),(24,106,69,114),(25,150,70,138),(26,108,61,116),(27,142,62,140),(28,110,63,118),(29,144,64,132),(30,102,65,120),(42,79,86,60),(44,71,88,52),(46,73,90,54),(48,75,82,56),(50,77,84,58),(51,160,80,125),(53,152,72,127),(55,154,74,129),(57,156,76,121),(59,158,78,123)], [(1,73,63,13),(2,14,64,74),(3,75,65,15),(4,16,66,76),(5,77,67,17),(6,18,68,78),(7,79,69,19),(8,20,70,80),(9,71,61,11),(10,12,62,72),(21,57,33,93),(22,94,34,58),(23,59,35,95),(24,96,36,60),(25,51,37,97),(26,98,38,52),(27,53,39,99),(28,100,40,54),(29,55,31,91),(30,92,32,56),(41,136,123,105),(42,106,124,137),(43,138,125,107),(44,108,126,139),(45,140,127,109),(46,110,128,131),(47,132,129,101),(48,102,130,133),(49,134,121,103),(50,104,122,135),(81,144,154,119),(82,120,155,145),(83,146,156,111),(84,112,157,147),(85,148,158,113),(86,114,159,149),(87,150,160,115),(88,116,151,141),(89,142,152,117),(90,118,153,143)], [(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)], [(1,10,13,72,63,62,73,12),(2,11,74,61,64,71,14,9),(3,8,15,80,65,70,75,20),(4,19,76,69,66,79,16,7),(5,6,17,78,67,68,77,18),(21,60,93,36,33,96,57,24),(22,23,58,95,34,35,94,59),(25,56,97,32,37,92,51,30),(26,29,52,91,38,31,98,55),(27,54,99,40,39,100,53,28),(41,147,105,157,123,112,136,84),(42,83,137,111,124,156,106,146),(43,145,107,155,125,120,138,82),(44,81,139,119,126,154,108,144),(45,143,109,153,127,118,140,90),(46,89,131,117,128,152,110,142),(47,141,101,151,129,116,132,88),(48,87,133,115,130,160,102,150),(49,149,103,159,121,114,134,86),(50,85,135,113,122,158,104,148)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 40 | 2 | ··· | 2 | 8 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | D4×D5 | D4.8D10 |
| kernel | C42.214D10 | C4×C5⋊2C8 | C4.D20 | C2×D4⋊D5 | C2×D4.D5 | C2×Q8⋊D5 | C2×C5⋊Q16 | C5×C4.4D4 | C5⋊2C8 | C2×C20 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 4 | 8 |
Matrix representation of C42.214D10 ►in GL6(𝔽41)
| 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 18 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 38 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 29 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 38 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 24 |
| 0 | 0 | 0 | 0 | 29 | 0 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,9,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,24,3,0,0,0,0,38,3,0,0,0,0,0,0,0,29,0,0,0,0,17,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,38,0,0,0,0,1,24,0,0,0,0,0,0,24,29,0,0,0,0,24,0] >;
C42.214D10 in GAP, Magma, Sage, TeX
C_4^2._{214}D_{10} % in TeX
G:=Group("C4^2.214D10"); // GroupNames label
G:=SmallGroup(320,686);
// by ID
G=gap.SmallGroup(320,686);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations