metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.65D10, C4.13(D4×D5), C5⋊2C8.7D4, C20.27(C2×D4), (C2×C20).83D4, C5⋊2(C8.2D4), C20⋊2Q8⋊18C2, (C2×D4).50D10, (C2×Q8).40D10, C4.4D4.7D5, C2.10(C20⋊D4), C10.19(C4⋊1D4), (C4×C20).109C22, (C2×C20).378C23, (D4×C10).66C22, C42.D5⋊10C2, (Q8×C10).58C22, C2.19(D4.9D10), C10.120(C8.C22), (C2×Dic10).112C22, (C2×C5⋊Q16)⋊14C2, (C2×D4.D5).7C2, (C2×C10).509(C2×D4), (C2×C4).63(C5⋊D4), (C5×C4.4D4).5C2, (C2×C4).478(C22×D5), C22.184(C2×C5⋊D4), (C2×C5⋊2C8).123C22, SmallGroup(320,687)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.65D10
G = < a,b,c,d | a4=b4=c10=1, d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 446 in 124 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, C10, C10 [×2], C10, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], Q16 [×4], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×3], C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C5⋊2C8 [×4], Dic10 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×C10, C8.2D4, C2×C5⋊2C8 [×2], C4⋊Dic5 [×2], D4.D5 [×4], C5⋊Q16 [×4], C4×C20, C5×C22⋊C4 [×2], C2×Dic10 [×2], D4×C10, Q8×C10, C42.D5, C20⋊2Q8, C2×D4.D5 [×2], C2×C5⋊Q16 [×2], C5×C4.4D4, C42.65D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C4⋊1D4, C8.C22 [×2], C5⋊D4 [×2], C22×D5, C8.2D4, D4×D5 [×2], C2×C5⋊D4, C20⋊D4, D4.9D10 [×2], C42.65D10
►On 160 points
Generators in S160
(1 45 13 53)(2 41 14 59)(3 47 15 55)(4 43 11 51)(5 49 12 57)(6 56 20 48)(7 52 16 44)(8 58 17 50)(9 54 18 46)(10 60 19 42)(21 64 33 69)(22 74 34 79)(23 66 35 61)(24 76 36 71)(25 68 37 63)(26 78 38 73)(27 70 39 65)(28 80 40 75)(29 62 31 67)(30 72 32 77)(81 121 91 143)(82 117 92 153)(83 123 93 145)(84 119 94 155)(85 125 95 147)(86 111 96 157)(87 127 97 149)(88 113 98 159)(89 129 99 141)(90 115 100 151)(101 142 139 130)(102 152 140 116)(103 144 131 122)(104 154 132 118)(105 146 133 124)(106 156 134 120)(107 148 135 126)(108 158 136 112)(109 150 137 128)(110 160 138 114)
(1 26 8 21)(2 22 9 27)(3 28 10 23)(4 24 6 29)(5 30 7 25)(11 36 20 31)(12 32 16 37)(13 38 17 33)(14 34 18 39)(15 40 19 35)(41 74 54 70)(42 61 55 75)(43 76 56 62)(44 63 57 77)(45 78 58 64)(46 65 59 79)(47 80 60 66)(48 67 51 71)(49 72 52 68)(50 69 53 73)(81 86 102 107)(82 108 103 87)(83 88 104 109)(84 110 105 89)(85 90 106 101)(91 96 140 135)(92 136 131 97)(93 98 132 137)(94 138 133 99)(95 100 134 139)(111 152 148 121)(112 122 149 153)(113 154 150 123)(114 124 141 155)(115 156 142 125)(116 126 143 157)(117 158 144 127)(118 128 145 159)(119 160 146 129)(120 130 147 151)
(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 85 26 90 8 106 21 101)(2 89 22 84 9 110 27 105)(3 83 28 88 10 104 23 109)(4 87 24 82 6 108 29 103)(5 81 30 86 7 102 25 107)(11 97 36 92 20 136 31 131)(12 91 32 96 16 140 37 135)(13 95 38 100 17 134 33 139)(14 99 34 94 18 138 39 133)(15 93 40 98 19 132 35 137)(41 160 74 146 54 129 70 119)(42 145 61 159 55 118 75 128)(43 158 76 144 56 127 62 117)(44 143 63 157 57 116 77 126)(45 156 78 142 58 125 64 115)(46 141 65 155 59 114 79 124)(47 154 80 150 60 123 66 113)(48 149 67 153 51 112 71 122)(49 152 72 148 52 121 68 111)(50 147 69 151 53 120 73 130)
G:=sub<Sym(160)| (1,45,13,53)(2,41,14,59)(3,47,15,55)(4,43,11,51)(5,49,12,57)(6,56,20,48)(7,52,16,44)(8,58,17,50)(9,54,18,46)(10,60,19,42)(21,64,33,69)(22,74,34,79)(23,66,35,61)(24,76,36,71)(25,68,37,63)(26,78,38,73)(27,70,39,65)(28,80,40,75)(29,62,31,67)(30,72,32,77)(81,121,91,143)(82,117,92,153)(83,123,93,145)(84,119,94,155)(85,125,95,147)(86,111,96,157)(87,127,97,149)(88,113,98,159)(89,129,99,141)(90,115,100,151)(101,142,139,130)(102,152,140,116)(103,144,131,122)(104,154,132,118)(105,146,133,124)(106,156,134,120)(107,148,135,126)(108,158,136,112)(109,150,137,128)(110,160,138,114), (1,26,8,21)(2,22,9,27)(3,28,10,23)(4,24,6,29)(5,30,7,25)(11,36,20,31)(12,32,16,37)(13,38,17,33)(14,34,18,39)(15,40,19,35)(41,74,54,70)(42,61,55,75)(43,76,56,62)(44,63,57,77)(45,78,58,64)(46,65,59,79)(47,80,60,66)(48,67,51,71)(49,72,52,68)(50,69,53,73)(81,86,102,107)(82,108,103,87)(83,88,104,109)(84,110,105,89)(85,90,106,101)(91,96,140,135)(92,136,131,97)(93,98,132,137)(94,138,133,99)(95,100,134,139)(111,152,148,121)(112,122,149,153)(113,154,150,123)(114,124,141,155)(115,156,142,125)(116,126,143,157)(117,158,144,127)(118,128,145,159)(119,160,146,129)(120,130,147,151), (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), (1,85,26,90,8,106,21,101)(2,89,22,84,9,110,27,105)(3,83,28,88,10,104,23,109)(4,87,24,82,6,108,29,103)(5,81,30,86,7,102,25,107)(11,97,36,92,20,136,31,131)(12,91,32,96,16,140,37,135)(13,95,38,100,17,134,33,139)(14,99,34,94,18,138,39,133)(15,93,40,98,19,132,35,137)(41,160,74,146,54,129,70,119)(42,145,61,159,55,118,75,128)(43,158,76,144,56,127,62,117)(44,143,63,157,57,116,77,126)(45,156,78,142,58,125,64,115)(46,141,65,155,59,114,79,124)(47,154,80,150,60,123,66,113)(48,149,67,153,51,112,71,122)(49,152,72,148,52,121,68,111)(50,147,69,151,53,120,73,130)>;
G:=Group( (1,45,13,53)(2,41,14,59)(3,47,15,55)(4,43,11,51)(5,49,12,57)(6,56,20,48)(7,52,16,44)(8,58,17,50)(9,54,18,46)(10,60,19,42)(21,64,33,69)(22,74,34,79)(23,66,35,61)(24,76,36,71)(25,68,37,63)(26,78,38,73)(27,70,39,65)(28,80,40,75)(29,62,31,67)(30,72,32,77)(81,121,91,143)(82,117,92,153)(83,123,93,145)(84,119,94,155)(85,125,95,147)(86,111,96,157)(87,127,97,149)(88,113,98,159)(89,129,99,141)(90,115,100,151)(101,142,139,130)(102,152,140,116)(103,144,131,122)(104,154,132,118)(105,146,133,124)(106,156,134,120)(107,148,135,126)(108,158,136,112)(109,150,137,128)(110,160,138,114), (1,26,8,21)(2,22,9,27)(3,28,10,23)(4,24,6,29)(5,30,7,25)(11,36,20,31)(12,32,16,37)(13,38,17,33)(14,34,18,39)(15,40,19,35)(41,74,54,70)(42,61,55,75)(43,76,56,62)(44,63,57,77)(45,78,58,64)(46,65,59,79)(47,80,60,66)(48,67,51,71)(49,72,52,68)(50,69,53,73)(81,86,102,107)(82,108,103,87)(83,88,104,109)(84,110,105,89)(85,90,106,101)(91,96,140,135)(92,136,131,97)(93,98,132,137)(94,138,133,99)(95,100,134,139)(111,152,148,121)(112,122,149,153)(113,154,150,123)(114,124,141,155)(115,156,142,125)(116,126,143,157)(117,158,144,127)(118,128,145,159)(119,160,146,129)(120,130,147,151), (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), (1,85,26,90,8,106,21,101)(2,89,22,84,9,110,27,105)(3,83,28,88,10,104,23,109)(4,87,24,82,6,108,29,103)(5,81,30,86,7,102,25,107)(11,97,36,92,20,136,31,131)(12,91,32,96,16,140,37,135)(13,95,38,100,17,134,33,139)(14,99,34,94,18,138,39,133)(15,93,40,98,19,132,35,137)(41,160,74,146,54,129,70,119)(42,145,61,159,55,118,75,128)(43,158,76,144,56,127,62,117)(44,143,63,157,57,116,77,126)(45,156,78,142,58,125,64,115)(46,141,65,155,59,114,79,124)(47,154,80,150,60,123,66,113)(48,149,67,153,51,112,71,122)(49,152,72,148,52,121,68,111)(50,147,69,151,53,120,73,130) );
G=PermutationGroup([(1,45,13,53),(2,41,14,59),(3,47,15,55),(4,43,11,51),(5,49,12,57),(6,56,20,48),(7,52,16,44),(8,58,17,50),(9,54,18,46),(10,60,19,42),(21,64,33,69),(22,74,34,79),(23,66,35,61),(24,76,36,71),(25,68,37,63),(26,78,38,73),(27,70,39,65),(28,80,40,75),(29,62,31,67),(30,72,32,77),(81,121,91,143),(82,117,92,153),(83,123,93,145),(84,119,94,155),(85,125,95,147),(86,111,96,157),(87,127,97,149),(88,113,98,159),(89,129,99,141),(90,115,100,151),(101,142,139,130),(102,152,140,116),(103,144,131,122),(104,154,132,118),(105,146,133,124),(106,156,134,120),(107,148,135,126),(108,158,136,112),(109,150,137,128),(110,160,138,114)], [(1,26,8,21),(2,22,9,27),(3,28,10,23),(4,24,6,29),(5,30,7,25),(11,36,20,31),(12,32,16,37),(13,38,17,33),(14,34,18,39),(15,40,19,35),(41,74,54,70),(42,61,55,75),(43,76,56,62),(44,63,57,77),(45,78,58,64),(46,65,59,79),(47,80,60,66),(48,67,51,71),(49,72,52,68),(50,69,53,73),(81,86,102,107),(82,108,103,87),(83,88,104,109),(84,110,105,89),(85,90,106,101),(91,96,140,135),(92,136,131,97),(93,98,132,137),(94,138,133,99),(95,100,134,139),(111,152,148,121),(112,122,149,153),(113,154,150,123),(114,124,141,155),(115,156,142,125),(116,126,143,157),(117,158,144,127),(118,128,145,159),(119,160,146,129),(120,130,147,151)], [(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)], [(1,85,26,90,8,106,21,101),(2,89,22,84,9,110,27,105),(3,83,28,88,10,104,23,109),(4,87,24,82,6,108,29,103),(5,81,30,86,7,102,25,107),(11,97,36,92,20,136,31,131),(12,91,32,96,16,140,37,135),(13,95,38,100,17,134,33,139),(14,99,34,94,18,138,39,133),(15,93,40,98,19,132,35,137),(41,160,74,146,54,129,70,119),(42,145,61,159,55,118,75,128),(43,158,76,144,56,127,62,117),(44,143,63,157,57,116,77,126),(45,156,78,142,58,125,64,115),(46,141,65,155,59,114,79,124),(47,154,80,150,60,123,66,113),(48,149,67,153,51,112,71,122),(49,152,72,148,52,121,68,111),(50,147,69,151,53,120,73,130)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 4 | 8 | 40 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C8.C22 | D4×D5 | D4.9D10 |
| kernel | C42.65D10 | C42.D5 | C20⋊2Q8 | C2×D4.D5 | C2×C5⋊Q16 | C5×C4.4D4 | C5⋊2C8 | C2×C20 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C2×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 8 |
Matrix representation of C42.65D10 ►in GL8(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 9 | 6 | 25 |
| 0 | 0 | 0 | 0 | 38 | 11 | 40 | 35 |
| 0 | 0 | 0 | 0 | 25 | 12 | 6 | 32 |
| 0 | 0 | 0 | 0 | 22 | 34 | 1 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 37 | 1 | 24 | 4 |
| 0 | 0 | 0 | 0 | 38 | 11 | 40 | 0 |
| 0 | 0 | 0 | 0 | 19 | 36 | 34 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 26 | 12 | 7 |
| 0 | 0 | 0 | 0 | 31 | 25 | 13 | 35 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 15 | 28 |
| 0 | 0 | 0 | 0 | 4 | 0 | 31 | 37 |
| 0 | 0 | 0 | 0 | 2 | 20 | 9 | 36 |
| 0 | 0 | 0 | 0 | 9 | 1 | 7 | 19 |
G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,5,38,25,22,0,0,0,0,9,11,12,34,0,0,0,0,6,40,6,1,0,0,0,0,25,35,32,19],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,37,38,19,0,0,0,0,0,1,11,36,0,0,0,0,34,24,40,34,0,0,0,0,1,4,0,0],[18,0,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,1,7,26,31,0,0,0,0,34,34,26,25,0,0,0,0,0,0,12,13,0,0,0,0,0,0,7,35],[0,0,18,0,0,0,0,0,0,0,0,23,0,0,0,0,16,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,13,4,2,9,0,0,0,0,0,0,20,1,0,0,0,0,15,31,9,7,0,0,0,0,28,37,36,19] >;
C42.65D10 in GAP, Magma, Sage, TeX
C_4^2._{65}D_{10} % in TeX
G:=Group("C4^2.65D10"); // GroupNames label
G:=SmallGroup(320,687);
// by ID
G=gap.SmallGroup(320,687);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,555,1123,297,136,12550]);
// 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*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^-1>;
// generators/relations