metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.70D10, C4⋊C4.76D10, (C2×C20).85D4, C42.C2⋊2D5, C20⋊4D4.8C2, D20⋊6C4⋊41C2, C20.71(C4○D4), (C2×C20).385C23, (C4×C20).115C22, C4.13(Q8⋊2D5), C42.D5⋊11C2, C2.22(D4⋊D10), C10.123(C8⋊C22), C10.55(C4.4D4), C2.8(C20.23D4), (C2×D20).107C22, C5⋊3(C42.29C22), (C5×C42.C2)⋊2C2, (C2×C10).516(C2×D4), (C2×C4).67(C5⋊D4), (C5×C4⋊C4).123C22, (C2×C4).483(C22×D5), C22.189(C2×C5⋊D4), (C2×C5⋊2C8).127C22, SmallGroup(320,694)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.70D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >
Subgroups: 590 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, C42.C2, C4⋊1D4, C5⋊2C8, D20, C2×C20, C2×C20, C2×C20, C22×D5, C42.29C22, C2×C5⋊2C8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×D20, C2×D20, C42.D5, D20⋊6C4, C20⋊4D4, C5×C42.C2, C42.70D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, C5⋊D4, C22×D5, C42.29C22, Q8⋊2D5, C2×C5⋊D4, C20.23D4, D4⋊D10, C42.70D10
►On 160 points
Generators in S160
(1 150 11 160)(2 74 12 64)(3 152 13 142)(4 76 14 66)(5 154 15 144)(6 78 16 68)(7 156 17 146)(8 80 18 70)(9 158 19 148)(10 62 20 72)(21 47 31 57)(22 114 32 104)(23 49 33 59)(24 116 34 106)(25 51 35 41)(26 118 36 108)(27 53 37 43)(28 120 38 110)(29 55 39 45)(30 102 40 112)(42 126 52 136)(44 128 54 138)(46 130 56 140)(48 132 58 122)(50 134 60 124)(61 86 71 96)(63 88 73 98)(65 90 75 100)(67 92 77 82)(69 94 79 84)(81 143 91 153)(83 145 93 155)(85 147 95 157)(87 149 97 159)(89 151 99 141)(101 139 111 129)(103 121 113 131)(105 123 115 133)(107 125 117 135)(109 127 119 137)
(1 131 98 21)(2 22 99 132)(3 133 100 23)(4 24 81 134)(5 135 82 25)(6 26 83 136)(7 137 84 27)(8 28 85 138)(9 139 86 29)(10 30 87 140)(11 121 88 31)(12 32 89 122)(13 123 90 33)(14 34 91 124)(15 125 92 35)(16 36 93 126)(17 127 94 37)(18 38 95 128)(19 129 96 39)(20 40 97 130)(41 144 117 77)(42 78 118 145)(43 146 119 79)(44 80 120 147)(45 148 101 61)(46 62 102 149)(47 150 103 63)(48 64 104 151)(49 152 105 65)(50 66 106 153)(51 154 107 67)(52 68 108 155)(53 156 109 69)(54 70 110 157)(55 158 111 71)(56 72 112 159)(57 160 113 73)(58 74 114 141)(59 142 115 75)(60 76 116 143)
(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 31 130 98 87 121 40)(2 39 122 86 99 129 32 9)(3 8 33 128 100 85 123 38)(4 37 124 84 81 127 34 7)(5 6 35 126 82 83 125 36)(11 20 21 140 88 97 131 30)(12 29 132 96 89 139 22 19)(13 18 23 138 90 95 133 28)(14 27 134 94 91 137 24 17)(15 16 25 136 92 93 135 26)(41 108 67 78 117 52 154 145)(42 144 155 51 118 77 68 107)(43 106 69 76 119 50 156 143)(44 142 157 49 120 75 70 105)(45 104 71 74 101 48 158 141)(46 160 159 47 102 73 72 103)(53 116 79 66 109 60 146 153)(54 152 147 59 110 65 80 115)(55 114 61 64 111 58 148 151)(56 150 149 57 112 63 62 113)
G:=sub<Sym(160)| (1,150,11,160)(2,74,12,64)(3,152,13,142)(4,76,14,66)(5,154,15,144)(6,78,16,68)(7,156,17,146)(8,80,18,70)(9,158,19,148)(10,62,20,72)(21,47,31,57)(22,114,32,104)(23,49,33,59)(24,116,34,106)(25,51,35,41)(26,118,36,108)(27,53,37,43)(28,120,38,110)(29,55,39,45)(30,102,40,112)(42,126,52,136)(44,128,54,138)(46,130,56,140)(48,132,58,122)(50,134,60,124)(61,86,71,96)(63,88,73,98)(65,90,75,100)(67,92,77,82)(69,94,79,84)(81,143,91,153)(83,145,93,155)(85,147,95,157)(87,149,97,159)(89,151,99,141)(101,139,111,129)(103,121,113,131)(105,123,115,133)(107,125,117,135)(109,127,119,137), (1,131,98,21)(2,22,99,132)(3,133,100,23)(4,24,81,134)(5,135,82,25)(6,26,83,136)(7,137,84,27)(8,28,85,138)(9,139,86,29)(10,30,87,140)(11,121,88,31)(12,32,89,122)(13,123,90,33)(14,34,91,124)(15,125,92,35)(16,36,93,126)(17,127,94,37)(18,38,95,128)(19,129,96,39)(20,40,97,130)(41,144,117,77)(42,78,118,145)(43,146,119,79)(44,80,120,147)(45,148,101,61)(46,62,102,149)(47,150,103,63)(48,64,104,151)(49,152,105,65)(50,66,106,153)(51,154,107,67)(52,68,108,155)(53,156,109,69)(54,70,110,157)(55,158,111,71)(56,72,112,159)(57,160,113,73)(58,74,114,141)(59,142,115,75)(60,76,116,143), (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), (1,10,31,130,98,87,121,40)(2,39,122,86,99,129,32,9)(3,8,33,128,100,85,123,38)(4,37,124,84,81,127,34,7)(5,6,35,126,82,83,125,36)(11,20,21,140,88,97,131,30)(12,29,132,96,89,139,22,19)(13,18,23,138,90,95,133,28)(14,27,134,94,91,137,24,17)(15,16,25,136,92,93,135,26)(41,108,67,78,117,52,154,145)(42,144,155,51,118,77,68,107)(43,106,69,76,119,50,156,143)(44,142,157,49,120,75,70,105)(45,104,71,74,101,48,158,141)(46,160,159,47,102,73,72,103)(53,116,79,66,109,60,146,153)(54,152,147,59,110,65,80,115)(55,114,61,64,111,58,148,151)(56,150,149,57,112,63,62,113)>;
G:=Group( (1,150,11,160)(2,74,12,64)(3,152,13,142)(4,76,14,66)(5,154,15,144)(6,78,16,68)(7,156,17,146)(8,80,18,70)(9,158,19,148)(10,62,20,72)(21,47,31,57)(22,114,32,104)(23,49,33,59)(24,116,34,106)(25,51,35,41)(26,118,36,108)(27,53,37,43)(28,120,38,110)(29,55,39,45)(30,102,40,112)(42,126,52,136)(44,128,54,138)(46,130,56,140)(48,132,58,122)(50,134,60,124)(61,86,71,96)(63,88,73,98)(65,90,75,100)(67,92,77,82)(69,94,79,84)(81,143,91,153)(83,145,93,155)(85,147,95,157)(87,149,97,159)(89,151,99,141)(101,139,111,129)(103,121,113,131)(105,123,115,133)(107,125,117,135)(109,127,119,137), (1,131,98,21)(2,22,99,132)(3,133,100,23)(4,24,81,134)(5,135,82,25)(6,26,83,136)(7,137,84,27)(8,28,85,138)(9,139,86,29)(10,30,87,140)(11,121,88,31)(12,32,89,122)(13,123,90,33)(14,34,91,124)(15,125,92,35)(16,36,93,126)(17,127,94,37)(18,38,95,128)(19,129,96,39)(20,40,97,130)(41,144,117,77)(42,78,118,145)(43,146,119,79)(44,80,120,147)(45,148,101,61)(46,62,102,149)(47,150,103,63)(48,64,104,151)(49,152,105,65)(50,66,106,153)(51,154,107,67)(52,68,108,155)(53,156,109,69)(54,70,110,157)(55,158,111,71)(56,72,112,159)(57,160,113,73)(58,74,114,141)(59,142,115,75)(60,76,116,143), (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), (1,10,31,130,98,87,121,40)(2,39,122,86,99,129,32,9)(3,8,33,128,100,85,123,38)(4,37,124,84,81,127,34,7)(5,6,35,126,82,83,125,36)(11,20,21,140,88,97,131,30)(12,29,132,96,89,139,22,19)(13,18,23,138,90,95,133,28)(14,27,134,94,91,137,24,17)(15,16,25,136,92,93,135,26)(41,108,67,78,117,52,154,145)(42,144,155,51,118,77,68,107)(43,106,69,76,119,50,156,143)(44,142,157,49,120,75,70,105)(45,104,71,74,101,48,158,141)(46,160,159,47,102,73,72,103)(53,116,79,66,109,60,146,153)(54,152,147,59,110,65,80,115)(55,114,61,64,111,58,148,151)(56,150,149,57,112,63,62,113) );
G=PermutationGroup([[(1,150,11,160),(2,74,12,64),(3,152,13,142),(4,76,14,66),(5,154,15,144),(6,78,16,68),(7,156,17,146),(8,80,18,70),(9,158,19,148),(10,62,20,72),(21,47,31,57),(22,114,32,104),(23,49,33,59),(24,116,34,106),(25,51,35,41),(26,118,36,108),(27,53,37,43),(28,120,38,110),(29,55,39,45),(30,102,40,112),(42,126,52,136),(44,128,54,138),(46,130,56,140),(48,132,58,122),(50,134,60,124),(61,86,71,96),(63,88,73,98),(65,90,75,100),(67,92,77,82),(69,94,79,84),(81,143,91,153),(83,145,93,155),(85,147,95,157),(87,149,97,159),(89,151,99,141),(101,139,111,129),(103,121,113,131),(105,123,115,133),(107,125,117,135),(109,127,119,137)], [(1,131,98,21),(2,22,99,132),(3,133,100,23),(4,24,81,134),(5,135,82,25),(6,26,83,136),(7,137,84,27),(8,28,85,138),(9,139,86,29),(10,30,87,140),(11,121,88,31),(12,32,89,122),(13,123,90,33),(14,34,91,124),(15,125,92,35),(16,36,93,126),(17,127,94,37),(18,38,95,128),(19,129,96,39),(20,40,97,130),(41,144,117,77),(42,78,118,145),(43,146,119,79),(44,80,120,147),(45,148,101,61),(46,62,102,149),(47,150,103,63),(48,64,104,151),(49,152,105,65),(50,66,106,153),(51,154,107,67),(52,68,108,155),(53,156,109,69),(54,70,110,157),(55,158,111,71),(56,72,112,159),(57,160,113,73),(58,74,114,141),(59,142,115,75),(60,76,116,143)], [(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)], [(1,10,31,130,98,87,121,40),(2,39,122,86,99,129,32,9),(3,8,33,128,100,85,123,38),(4,37,124,84,81,127,34,7),(5,6,35,126,82,83,125,36),(11,20,21,140,88,97,131,30),(12,29,132,96,89,139,22,19),(13,18,23,138,90,95,133,28),(14,27,134,94,91,137,24,17),(15,16,25,136,92,93,135,26),(41,108,67,78,117,52,154,145),(42,144,155,51,118,77,68,107),(43,106,69,76,119,50,156,143),(44,142,157,49,120,75,70,105),(45,104,71,74,101,48,158,141),(46,160,159,47,102,73,72,103),(53,116,79,66,109,60,146,153),(54,152,147,59,110,65,80,115),(55,114,61,64,111,58,148,151),(56,150,149,57,112,63,62,113)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | C8⋊C22 | Q8⋊2D5 | D4⋊D10 |
| kernel | C42.70D10 | C42.D5 | D20⋊6C4 | C20⋊4D4 | C5×C42.C2 | C2×C20 | C42.C2 | C20 | C42 | C4⋊C4 | C2×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of C42.70D10 ►in GL6(𝔽41)
| 37 | 24 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 22 | 18 |
| 0 | 0 | 9 | 11 | 23 | 19 |
| 0 | 0 | 30 | 32 | 11 | 9 |
| 0 | 0 | 9 | 11 | 32 | 30 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 39 | 0 |
| 0 | 0 | 0 | 1 | 0 | 39 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 36 | 12 | 0 | 0 | 0 | 0 |
| 32 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 5 | 24 |
| 0 | 0 | 35 | 5 | 17 | 1 |
| 0 | 0 | 29 | 18 | 35 | 35 |
| 0 | 0 | 23 | 26 | 6 | 36 |
| 36 | 30 | 0 | 0 | 0 | 0 |
| 32 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 11 | 17 | 36 |
| 0 | 0 | 6 | 11 | 40 | 24 |
| 0 | 0 | 12 | 23 | 6 | 6 |
| 0 | 0 | 21 | 29 | 5 | 35 |
G:=sub<GL(6,GF(41))| [37,1,0,0,0,0,24,4,0,0,0,0,0,0,30,9,30,9,0,0,32,11,32,11,0,0,22,23,11,32,0,0,18,19,9,30],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,39,0,40,0,0,0,0,39,0,40],[36,32,0,0,0,0,12,5,0,0,0,0,0,0,6,35,29,23,0,0,6,5,18,26,0,0,5,17,35,6,0,0,24,1,35,36],[36,32,0,0,0,0,30,5,0,0,0,0,0,0,30,6,12,21,0,0,11,11,23,29,0,0,17,40,6,5,0,0,36,24,6,35] >;
C42.70D10 in GAP, Magma, Sage, TeX
C_4^2._{70}D_{10} % in TeX
G:=Group("C4^2.70D10"); // GroupNames label
G:=SmallGroup(320,694);
// by ID
G=gap.SmallGroup(320,694);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,555,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^9>;
// generators/relations