metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.47D10, C5⋊2C8⋊28D4, C5⋊9(C8⋊9D4), (C4×D4).5D5, C20⋊3C8⋊21C2, (D4×C20).6C2, C4.215(D4×D5), C4⋊C4.7Dic5, C2.8(D4×Dic5), (D4×C10).28C4, C20.374(C2×D4), C10.121(C4×D4), (C2×C10)⋊7M4(2), (C2×D4).5Dic5, C10.63(C8○D4), (C4×C20).82C22, C42.D5⋊4C2, C22⋊C4.4Dic5, C20.307(C4○D4), C20.55D4⋊25C2, (C2×C20).849C23, (C22×C4).310D10, C10.74(C2×M4(2)), C2.6(D4.Dic5), C22⋊1(C4.Dic5), C4.134(D4⋊2D5), C23.17(C2×Dic5), (C22×C20).99C22, C22.45(C22×Dic5), (C5×C4⋊C4).24C4, (C2×C4.Dic5)⋊4C2, (C2×C20).336(C2×C4), C2.8(C2×C4.Dic5), (C22×C5⋊2C8)⋊19C2, (C5×C22⋊C4).13C4, (C2×C4).34(C2×Dic5), (C2×C4).791(C22×D5), (C22×C10).128(C2×C4), (C2×C10).287(C22×C4), (C2×C5⋊2C8).204C22, SmallGroup(320,638)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.47D10
G = < a,b,c,d | a4=b4=c10=1, d2=b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 286 in 124 conjugacy classes, 61 normal (55 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C5×D4, C22×C10, C8⋊9D4, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C42.D5, C20⋊3C8, C20.55D4, C22×C5⋊2C8, C2×C4.Dic5, D4×C20, C42.47D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C2×M4(2), C8○D4, C2×Dic5, C22×D5, C8⋊9D4, C4.Dic5, D4×D5, D4⋊2D5, C22×Dic5, C2×C4.Dic5, D4×Dic5, D4.Dic5, C42.47D10
►On 160 points
Generators in S160
(1 52 57 6)(2 7 58 53)(3 54 59 8)(4 9 60 55)(5 56 51 10)(11 16 29 24)(12 25 30 17)(13 18 21 26)(14 27 22 19)(15 20 23 28)(31 36 64 69)(32 70 65 37)(33 38 66 61)(34 62 67 39)(35 40 68 63)(41 124 158 107)(42 108 159 125)(43 126 160 109)(44 110 151 127)(45 128 152 101)(46 102 153 129)(47 130 154 103)(48 104 155 121)(49 122 156 105)(50 106 157 123)(71 84 89 76)(72 77 90 85)(73 86 81 78)(74 79 82 87)(75 88 83 80)(91 120 133 142)(92 143 134 111)(93 112 135 144)(94 145 136 113)(95 114 137 146)(96 147 138 115)(97 116 139 148)(98 149 140 117)(99 118 131 150)(100 141 132 119)
(1 73 63 24)(2 74 64 25)(3 75 65 26)(4 76 66 27)(5 77 67 28)(6 78 68 29)(7 79 69 30)(8 80 70 21)(9 71 61 22)(10 72 62 23)(11 52 86 35)(12 53 87 36)(13 54 88 37)(14 55 89 38)(15 56 90 39)(16 57 81 40)(17 58 82 31)(18 59 83 32)(19 60 84 33)(20 51 85 34)(41 116 129 134)(42 117 130 135)(43 118 121 136)(44 119 122 137)(45 120 123 138)(46 111 124 139)(47 112 125 140)(48 113 126 131)(49 114 127 132)(50 115 128 133)(91 157 147 101)(92 158 148 102)(93 159 149 103)(94 160 150 104)(95 151 141 105)(96 152 142 106)(97 153 143 107)(98 154 144 108)(99 155 145 109)(100 156 146 110)
(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)
(1 156 73 146 63 110 24 100)(2 155 74 145 64 109 25 99)(3 154 75 144 65 108 26 98)(4 153 76 143 66 107 27 97)(5 152 77 142 67 106 28 96)(6 151 78 141 68 105 29 95)(7 160 79 150 69 104 30 94)(8 159 80 149 70 103 21 93)(9 158 71 148 61 102 22 92)(10 157 72 147 62 101 23 91)(11 137 52 44 86 119 35 122)(12 136 53 43 87 118 36 121)(13 135 54 42 88 117 37 130)(14 134 55 41 89 116 38 129)(15 133 56 50 90 115 39 128)(16 132 57 49 81 114 40 127)(17 131 58 48 82 113 31 126)(18 140 59 47 83 112 32 125)(19 139 60 46 84 111 33 124)(20 138 51 45 85 120 34 123)
G:=sub<Sym(160)| (1,52,57,6)(2,7,58,53)(3,54,59,8)(4,9,60,55)(5,56,51,10)(11,16,29,24)(12,25,30,17)(13,18,21,26)(14,27,22,19)(15,20,23,28)(31,36,64,69)(32,70,65,37)(33,38,66,61)(34,62,67,39)(35,40,68,63)(41,124,158,107)(42,108,159,125)(43,126,160,109)(44,110,151,127)(45,128,152,101)(46,102,153,129)(47,130,154,103)(48,104,155,121)(49,122,156,105)(50,106,157,123)(71,84,89,76)(72,77,90,85)(73,86,81,78)(74,79,82,87)(75,88,83,80)(91,120,133,142)(92,143,134,111)(93,112,135,144)(94,145,136,113)(95,114,137,146)(96,147,138,115)(97,116,139,148)(98,149,140,117)(99,118,131,150)(100,141,132,119), (1,73,63,24)(2,74,64,25)(3,75,65,26)(4,76,66,27)(5,77,67,28)(6,78,68,29)(7,79,69,30)(8,80,70,21)(9,71,61,22)(10,72,62,23)(11,52,86,35)(12,53,87,36)(13,54,88,37)(14,55,89,38)(15,56,90,39)(16,57,81,40)(17,58,82,31)(18,59,83,32)(19,60,84,33)(20,51,85,34)(41,116,129,134)(42,117,130,135)(43,118,121,136)(44,119,122,137)(45,120,123,138)(46,111,124,139)(47,112,125,140)(48,113,126,131)(49,114,127,132)(50,115,128,133)(91,157,147,101)(92,158,148,102)(93,159,149,103)(94,160,150,104)(95,151,141,105)(96,152,142,106)(97,153,143,107)(98,154,144,108)(99,155,145,109)(100,156,146,110), (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), (1,156,73,146,63,110,24,100)(2,155,74,145,64,109,25,99)(3,154,75,144,65,108,26,98)(4,153,76,143,66,107,27,97)(5,152,77,142,67,106,28,96)(6,151,78,141,68,105,29,95)(7,160,79,150,69,104,30,94)(8,159,80,149,70,103,21,93)(9,158,71,148,61,102,22,92)(10,157,72,147,62,101,23,91)(11,137,52,44,86,119,35,122)(12,136,53,43,87,118,36,121)(13,135,54,42,88,117,37,130)(14,134,55,41,89,116,38,129)(15,133,56,50,90,115,39,128)(16,132,57,49,81,114,40,127)(17,131,58,48,82,113,31,126)(18,140,59,47,83,112,32,125)(19,139,60,46,84,111,33,124)(20,138,51,45,85,120,34,123)>;
G:=Group( (1,52,57,6)(2,7,58,53)(3,54,59,8)(4,9,60,55)(5,56,51,10)(11,16,29,24)(12,25,30,17)(13,18,21,26)(14,27,22,19)(15,20,23,28)(31,36,64,69)(32,70,65,37)(33,38,66,61)(34,62,67,39)(35,40,68,63)(41,124,158,107)(42,108,159,125)(43,126,160,109)(44,110,151,127)(45,128,152,101)(46,102,153,129)(47,130,154,103)(48,104,155,121)(49,122,156,105)(50,106,157,123)(71,84,89,76)(72,77,90,85)(73,86,81,78)(74,79,82,87)(75,88,83,80)(91,120,133,142)(92,143,134,111)(93,112,135,144)(94,145,136,113)(95,114,137,146)(96,147,138,115)(97,116,139,148)(98,149,140,117)(99,118,131,150)(100,141,132,119), (1,73,63,24)(2,74,64,25)(3,75,65,26)(4,76,66,27)(5,77,67,28)(6,78,68,29)(7,79,69,30)(8,80,70,21)(9,71,61,22)(10,72,62,23)(11,52,86,35)(12,53,87,36)(13,54,88,37)(14,55,89,38)(15,56,90,39)(16,57,81,40)(17,58,82,31)(18,59,83,32)(19,60,84,33)(20,51,85,34)(41,116,129,134)(42,117,130,135)(43,118,121,136)(44,119,122,137)(45,120,123,138)(46,111,124,139)(47,112,125,140)(48,113,126,131)(49,114,127,132)(50,115,128,133)(91,157,147,101)(92,158,148,102)(93,159,149,103)(94,160,150,104)(95,151,141,105)(96,152,142,106)(97,153,143,107)(98,154,144,108)(99,155,145,109)(100,156,146,110), (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), (1,156,73,146,63,110,24,100)(2,155,74,145,64,109,25,99)(3,154,75,144,65,108,26,98)(4,153,76,143,66,107,27,97)(5,152,77,142,67,106,28,96)(6,151,78,141,68,105,29,95)(7,160,79,150,69,104,30,94)(8,159,80,149,70,103,21,93)(9,158,71,148,61,102,22,92)(10,157,72,147,62,101,23,91)(11,137,52,44,86,119,35,122)(12,136,53,43,87,118,36,121)(13,135,54,42,88,117,37,130)(14,134,55,41,89,116,38,129)(15,133,56,50,90,115,39,128)(16,132,57,49,81,114,40,127)(17,131,58,48,82,113,31,126)(18,140,59,47,83,112,32,125)(19,139,60,46,84,111,33,124)(20,138,51,45,85,120,34,123) );
G=PermutationGroup([[(1,52,57,6),(2,7,58,53),(3,54,59,8),(4,9,60,55),(5,56,51,10),(11,16,29,24),(12,25,30,17),(13,18,21,26),(14,27,22,19),(15,20,23,28),(31,36,64,69),(32,70,65,37),(33,38,66,61),(34,62,67,39),(35,40,68,63),(41,124,158,107),(42,108,159,125),(43,126,160,109),(44,110,151,127),(45,128,152,101),(46,102,153,129),(47,130,154,103),(48,104,155,121),(49,122,156,105),(50,106,157,123),(71,84,89,76),(72,77,90,85),(73,86,81,78),(74,79,82,87),(75,88,83,80),(91,120,133,142),(92,143,134,111),(93,112,135,144),(94,145,136,113),(95,114,137,146),(96,147,138,115),(97,116,139,148),(98,149,140,117),(99,118,131,150),(100,141,132,119)], [(1,73,63,24),(2,74,64,25),(3,75,65,26),(4,76,66,27),(5,77,67,28),(6,78,68,29),(7,79,69,30),(8,80,70,21),(9,71,61,22),(10,72,62,23),(11,52,86,35),(12,53,87,36),(13,54,88,37),(14,55,89,38),(15,56,90,39),(16,57,81,40),(17,58,82,31),(18,59,83,32),(19,60,84,33),(20,51,85,34),(41,116,129,134),(42,117,130,135),(43,118,121,136),(44,119,122,137),(45,120,123,138),(46,111,124,139),(47,112,125,140),(48,113,126,131),(49,114,127,132),(50,115,128,133),(91,157,147,101),(92,158,148,102),(93,159,149,103),(94,160,150,104),(95,151,141,105),(96,152,142,106),(97,153,143,107),(98,154,144,108),(99,155,145,109),(100,156,146,110)], [(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)], [(1,156,73,146,63,110,24,100),(2,155,74,145,64,109,25,99),(3,154,75,144,65,108,26,98),(4,153,76,143,66,107,27,97),(5,152,77,142,67,106,28,96),(6,151,78,141,68,105,29,95),(7,160,79,150,69,104,30,94),(8,159,80,149,70,103,21,93),(9,158,71,148,61,102,22,92),(10,157,72,147,62,101,23,91),(11,137,52,44,86,119,35,122),(12,136,53,43,87,118,36,121),(13,135,54,42,88,117,37,130),(14,134,55,41,89,116,38,129),(15,133,56,50,90,115,39,128),(16,132,57,49,81,114,40,127),(17,131,58,48,82,113,31,126),(18,140,59,47,83,112,32,125),(19,139,60,46,84,111,33,124),(20,138,51,45,85,120,34,123)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | C4○D4 | M4(2) | D10 | Dic5 | Dic5 | D10 | Dic5 | C8○D4 | C4.Dic5 | D4×D5 | D4⋊2D5 | D4.Dic5 |
| kernel | C42.47D10 | C42.D5 | C20⋊3C8 | C20.55D4 | C22×C5⋊2C8 | C2×C4.Dic5 | D4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | D4×C10 | C5⋊2C8 | C4×D4 | C20 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of C42.47D10 ►in GL4(𝔽41) generated by
| 1 | 39 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 39 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 25 |
| 1 | 39 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 25 |
| 0 | 0 | 39 | 0 |
G:=sub<GL(4,GF(41))| [1,1,0,0,39,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,39,40,0,0,0,0,23,0,0,0,0,25],[1,0,0,0,39,40,0,0,0,0,0,39,0,0,25,0] >;
C42.47D10 in GAP, Magma, Sage, TeX
C_4^2._{47}D_{10} % in TeX
G:=Group("C4^2.47D10"); // GroupNames label
G:=SmallGroup(320,638);
// by ID
G=gap.SmallGroup(320,638);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations