metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.36D10, Dic10.18D4, C4⋊C8⋊6D5, (C2×C4).39D20, C4.132(D4×D5), (C2×Dic20)⋊7C2, C20.341(C2×D4), (C2×C8).131D10, (C2×C20).245D4, C5⋊3(Q8.D4), (C4×Dic10)⋊17C2, D20⋊5C4.2C2, C10.13(C4○D8), (C2×C40).24C22, (C4×C20).71C22, C4.D20.7C2, C20.330(C4○D4), C20.44D4⋊13C2, C2.13(C4⋊D20), C10.40(C4⋊D4), (C2×C20).755C23, C4.46(Q8⋊2D5), (C2×D20).18C22, C22.118(C2×D20), C2.15(D40⋊7C2), C2.18(C8.D10), C10.15(C8.C22), C4⋊Dic5.275C22, (C2×Dic10).220C22, (C5×C4⋊C8)⋊8C2, (C2×C40⋊C2).6C2, (C2×C10).138(C2×D4), (C2×C4).700(C22×D5), SmallGroup(320,472)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.36D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=b2, ab=ba, cac-1=a-1b2, ad=da, bc=cb, dbd-1=b-1, dcd-1=bc9 >
Subgroups: 518 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C5, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×5], C23, D5, C10 [×3], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×3], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C40 [×2], Dic10 [×2], Dic10 [×3], D20 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, Q8.D4, C40⋊C2 [×2], Dic20 [×2], C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×Dic10 [×2], C2×D20, C20.44D4, D20⋊5C4, C5×C4⋊C8, C4×Dic10, C4.D20, C2×C40⋊C2, C2×Dic20, C42.36D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8.C22, D20 [×2], C22×D5, Q8.D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D40⋊7C2, C8.D10, C42.36D10
►On 160 points
Generators in S160
(1 44 139 118)(2 99 140 65)(3 46 141 120)(4 101 142 67)(5 48 143 82)(6 103 144 69)(7 50 145 84)(8 105 146 71)(9 52 147 86)(10 107 148 73)(11 54 149 88)(12 109 150 75)(13 56 151 90)(14 111 152 77)(15 58 153 92)(16 113 154 79)(17 60 155 94)(18 115 156 41)(19 62 157 96)(20 117 158 43)(21 64 159 98)(22 119 160 45)(23 66 121 100)(24 81 122 47)(25 68 123 102)(26 83 124 49)(27 70 125 104)(28 85 126 51)(29 72 127 106)(30 87 128 53)(31 74 129 108)(32 89 130 55)(33 76 131 110)(34 91 132 57)(35 78 133 112)(36 93 134 59)(37 80 135 114)(38 95 136 61)(39 42 137 116)(40 97 138 63)
(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 21 30)(2 29 22 9)(3 8 23 28)(4 27 24 7)(5 6 25 26)(11 40 31 20)(12 19 32 39)(13 38 33 18)(14 17 34 37)(15 36 35 16)(41 90 61 110)(42 109 62 89)(43 88 63 108)(44 107 64 87)(45 86 65 106)(46 105 66 85)(47 84 67 104)(48 103 68 83)(49 82 69 102)(50 101 70 81)(51 120 71 100)(52 99 72 119)(53 118 73 98)(54 97 74 117)(55 116 75 96)(56 95 76 115)(57 114 77 94)(58 93 78 113)(59 112 79 92)(60 91 80 111)(121 126 141 146)(122 145 142 125)(123 124 143 144)(127 160 147 140)(128 139 148 159)(129 158 149 138)(130 137 150 157)(131 156 151 136)(132 135 152 155)(133 154 153 134)
G:=sub<Sym(160)| (1,44,139,118)(2,99,140,65)(3,46,141,120)(4,101,142,67)(5,48,143,82)(6,103,144,69)(7,50,145,84)(8,105,146,71)(9,52,147,86)(10,107,148,73)(11,54,149,88)(12,109,150,75)(13,56,151,90)(14,111,152,77)(15,58,153,92)(16,113,154,79)(17,60,155,94)(18,115,156,41)(19,62,157,96)(20,117,158,43)(21,64,159,98)(22,119,160,45)(23,66,121,100)(24,81,122,47)(25,68,123,102)(26,83,124,49)(27,70,125,104)(28,85,126,51)(29,72,127,106)(30,87,128,53)(31,74,129,108)(32,89,130,55)(33,76,131,110)(34,91,132,57)(35,78,133,112)(36,93,134,59)(37,80,135,114)(38,95,136,61)(39,42,137,116)(40,97,138,63), (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), (1,10,21,30)(2,29,22,9)(3,8,23,28)(4,27,24,7)(5,6,25,26)(11,40,31,20)(12,19,32,39)(13,38,33,18)(14,17,34,37)(15,36,35,16)(41,90,61,110)(42,109,62,89)(43,88,63,108)(44,107,64,87)(45,86,65,106)(46,105,66,85)(47,84,67,104)(48,103,68,83)(49,82,69,102)(50,101,70,81)(51,120,71,100)(52,99,72,119)(53,118,73,98)(54,97,74,117)(55,116,75,96)(56,95,76,115)(57,114,77,94)(58,93,78,113)(59,112,79,92)(60,91,80,111)(121,126,141,146)(122,145,142,125)(123,124,143,144)(127,160,147,140)(128,139,148,159)(129,158,149,138)(130,137,150,157)(131,156,151,136)(132,135,152,155)(133,154,153,134)>;
G:=Group( (1,44,139,118)(2,99,140,65)(3,46,141,120)(4,101,142,67)(5,48,143,82)(6,103,144,69)(7,50,145,84)(8,105,146,71)(9,52,147,86)(10,107,148,73)(11,54,149,88)(12,109,150,75)(13,56,151,90)(14,111,152,77)(15,58,153,92)(16,113,154,79)(17,60,155,94)(18,115,156,41)(19,62,157,96)(20,117,158,43)(21,64,159,98)(22,119,160,45)(23,66,121,100)(24,81,122,47)(25,68,123,102)(26,83,124,49)(27,70,125,104)(28,85,126,51)(29,72,127,106)(30,87,128,53)(31,74,129,108)(32,89,130,55)(33,76,131,110)(34,91,132,57)(35,78,133,112)(36,93,134,59)(37,80,135,114)(38,95,136,61)(39,42,137,116)(40,97,138,63), (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), (1,10,21,30)(2,29,22,9)(3,8,23,28)(4,27,24,7)(5,6,25,26)(11,40,31,20)(12,19,32,39)(13,38,33,18)(14,17,34,37)(15,36,35,16)(41,90,61,110)(42,109,62,89)(43,88,63,108)(44,107,64,87)(45,86,65,106)(46,105,66,85)(47,84,67,104)(48,103,68,83)(49,82,69,102)(50,101,70,81)(51,120,71,100)(52,99,72,119)(53,118,73,98)(54,97,74,117)(55,116,75,96)(56,95,76,115)(57,114,77,94)(58,93,78,113)(59,112,79,92)(60,91,80,111)(121,126,141,146)(122,145,142,125)(123,124,143,144)(127,160,147,140)(128,139,148,159)(129,158,149,138)(130,137,150,157)(131,156,151,136)(132,135,152,155)(133,154,153,134) );
G=PermutationGroup([(1,44,139,118),(2,99,140,65),(3,46,141,120),(4,101,142,67),(5,48,143,82),(6,103,144,69),(7,50,145,84),(8,105,146,71),(9,52,147,86),(10,107,148,73),(11,54,149,88),(12,109,150,75),(13,56,151,90),(14,111,152,77),(15,58,153,92),(16,113,154,79),(17,60,155,94),(18,115,156,41),(19,62,157,96),(20,117,158,43),(21,64,159,98),(22,119,160,45),(23,66,121,100),(24,81,122,47),(25,68,123,102),(26,83,124,49),(27,70,125,104),(28,85,126,51),(29,72,127,106),(30,87,128,53),(31,74,129,108),(32,89,130,55),(33,76,131,110),(34,91,132,57),(35,78,133,112),(36,93,134,59),(37,80,135,114),(38,95,136,61),(39,42,137,116),(40,97,138,63)], [(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)], [(1,10,21,30),(2,29,22,9),(3,8,23,28),(4,27,24,7),(5,6,25,26),(11,40,31,20),(12,19,32,39),(13,38,33,18),(14,17,34,37),(15,36,35,16),(41,90,61,110),(42,109,62,89),(43,88,63,108),(44,107,64,87),(45,86,65,106),(46,105,66,85),(47,84,67,104),(48,103,68,83),(49,82,69,102),(50,101,70,81),(51,120,71,100),(52,99,72,119),(53,118,73,98),(54,97,74,117),(55,116,75,96),(56,95,76,115),(57,114,77,94),(58,93,78,113),(59,112,79,92),(60,91,80,111),(121,126,141,146),(122,145,142,125),(123,124,143,144),(127,160,147,140),(128,139,148,159),(129,158,149,138),(130,137,150,157),(131,156,151,136),(132,135,152,155),(133,154,153,134)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 40 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | D20 | D40⋊7C2 | C8.C22 | D4×D5 | Q8⋊2D5 | C8.D10 |
| kernel | C42.36D10 | C20.44D4 | D20⋊5C4 | C5×C4⋊C8 | C4×Dic10 | C4.D20 | C2×C40⋊C2 | C2×Dic20 | Dic10 | C2×C20 | C4⋊C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C2 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of C42.36D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 33 | 5 |
| 0 | 0 | 28 | 8 |
| 39 | 32 | 0 | 0 |
| 37 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 23 | 0 | 0 |
| 33 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 5 | 1 |
| 2 | 23 | 0 | 0 |
| 39 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,33,28,0,0,5,8],[39,37,0,0,32,2,0,0,0,0,1,0,0,0,0,1],[31,33,0,0,23,39,0,0,0,0,40,5,0,0,0,1],[2,39,0,0,23,39,0,0,0,0,40,0,0,0,0,40] >;
C42.36D10 in GAP, Magma, Sage, TeX
C_4^2._{36}D_{10} % in TeX
G:=Group("C4^2.36D10"); // GroupNames label
G:=SmallGroup(320,472);
// by ID
G=gap.SmallGroup(320,472);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations