metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.151D10, C10.292- 1+4, C42.C2⋊7D5, C4⋊C4.112D10, D10⋊Q8⋊36C2, C42⋊D5⋊37C2, (C2×C20).89C23, D10.38(C4○D4), Dic5⋊3Q8⋊36C2, D20⋊8C4.12C2, (C2×C10).237C24, (C4×C20).240C22, Dic5.46(C4○D4), Dic5.Q8⋊34C2, D10.13D4.2C2, (C2×D20).171C22, C4⋊Dic5.242C22, C22.258(C23×D5), C5⋊9(C22.46C24), (C4×Dic5).235C22, (C2×Dic5).269C23, C10.D4.53C22, (C22×D5).232C23, D10⋊C4.137C22, C2.30(Q8.10D10), (C2×Dic10).187C22, (D5×C4⋊C4)⋊37C2, C2.88(D5×C4○D4), C4⋊C4⋊7D5⋊36C2, C4⋊C4⋊D5⋊35C2, C10.199(C2×C4○D4), (C5×C42.C2)⋊10C2, (C2×C4×D5).136C22, (C5×C4⋊C4).192C22, (C2×C4).204(C22×D5), SmallGroup(320,1365)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.151D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c9 >
Subgroups: 710 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C22.46C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C42⋊D5, Dic5⋊3Q8, Dic5.Q8, D5×C4⋊C4, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4, D10⋊Q8, C4⋊C4⋊D5, C5×C42.C2, C42.151D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, C23×D5, Q8.10D10, D5×C4○D4, C42.151D10
►On 160 points
Generators in S160
(1 106 11 116)(2 71 12 61)(3 108 13 118)(4 73 14 63)(5 110 15 120)(6 75 16 65)(7 112 17 102)(8 77 18 67)(9 114 19 104)(10 79 20 69)(21 86 31 96)(22 131 32 121)(23 88 33 98)(24 133 34 123)(25 90 35 100)(26 135 36 125)(27 92 37 82)(28 137 38 127)(29 94 39 84)(30 139 40 129)(41 64 51 74)(42 101 52 111)(43 66 53 76)(44 103 54 113)(45 68 55 78)(46 105 56 115)(47 70 57 80)(48 107 58 117)(49 72 59 62)(50 109 60 119)(81 153 91 143)(83 155 93 145)(85 157 95 147)(87 159 97 149)(89 141 99 151)(122 150 132 160)(124 152 134 142)(126 154 136 144)(128 156 138 146)(130 158 140 148)
(1 35 47 142)(2 26 48 153)(3 37 49 144)(4 28 50 155)(5 39 51 146)(6 30 52 157)(7 21 53 148)(8 32 54 159)(9 23 55 150)(10 34 56 141)(11 25 57 152)(12 36 58 143)(13 27 59 154)(14 38 60 145)(15 29 41 156)(16 40 42 147)(17 31 43 158)(18 22 44 149)(19 33 45 160)(20 24 46 151)(61 125 117 81)(62 136 118 92)(63 127 119 83)(64 138 120 94)(65 129 101 85)(66 140 102 96)(67 131 103 87)(68 122 104 98)(69 133 105 89)(70 124 106 100)(71 135 107 91)(72 126 108 82)(73 137 109 93)(74 128 110 84)(75 139 111 95)(76 130 112 86)(77 121 113 97)(78 132 114 88)(79 123 115 99)(80 134 116 90)
(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 84 11 94)(2 93 12 83)(3 82 13 92)(4 91 14 81)(5 100 15 90)(6 89 16 99)(7 98 17 88)(8 87 18 97)(9 96 19 86)(10 85 20 95)(21 68 31 78)(22 77 32 67)(23 66 33 76)(24 75 34 65)(25 64 35 74)(26 73 36 63)(27 62 37 72)(28 71 38 61)(29 80 39 70)(30 69 40 79)(41 134 51 124)(42 123 52 133)(43 132 53 122)(44 121 54 131)(45 130 55 140)(46 139 56 129)(47 128 57 138)(48 137 58 127)(49 126 59 136)(50 135 60 125)(101 151 111 141)(102 160 112 150)(103 149 113 159)(104 158 114 148)(105 147 115 157)(106 156 116 146)(107 145 117 155)(108 154 118 144)(109 143 119 153)(110 152 120 142)
G:=sub<Sym(160)| (1,106,11,116)(2,71,12,61)(3,108,13,118)(4,73,14,63)(5,110,15,120)(6,75,16,65)(7,112,17,102)(8,77,18,67)(9,114,19,104)(10,79,20,69)(21,86,31,96)(22,131,32,121)(23,88,33,98)(24,133,34,123)(25,90,35,100)(26,135,36,125)(27,92,37,82)(28,137,38,127)(29,94,39,84)(30,139,40,129)(41,64,51,74)(42,101,52,111)(43,66,53,76)(44,103,54,113)(45,68,55,78)(46,105,56,115)(47,70,57,80)(48,107,58,117)(49,72,59,62)(50,109,60,119)(81,153,91,143)(83,155,93,145)(85,157,95,147)(87,159,97,149)(89,141,99,151)(122,150,132,160)(124,152,134,142)(126,154,136,144)(128,156,138,146)(130,158,140,148), (1,35,47,142)(2,26,48,153)(3,37,49,144)(4,28,50,155)(5,39,51,146)(6,30,52,157)(7,21,53,148)(8,32,54,159)(9,23,55,150)(10,34,56,141)(11,25,57,152)(12,36,58,143)(13,27,59,154)(14,38,60,145)(15,29,41,156)(16,40,42,147)(17,31,43,158)(18,22,44,149)(19,33,45,160)(20,24,46,151)(61,125,117,81)(62,136,118,92)(63,127,119,83)(64,138,120,94)(65,129,101,85)(66,140,102,96)(67,131,103,87)(68,122,104,98)(69,133,105,89)(70,124,106,100)(71,135,107,91)(72,126,108,82)(73,137,109,93)(74,128,110,84)(75,139,111,95)(76,130,112,86)(77,121,113,97)(78,132,114,88)(79,123,115,99)(80,134,116,90), (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), (1,84,11,94)(2,93,12,83)(3,82,13,92)(4,91,14,81)(5,100,15,90)(6,89,16,99)(7,98,17,88)(8,87,18,97)(9,96,19,86)(10,85,20,95)(21,68,31,78)(22,77,32,67)(23,66,33,76)(24,75,34,65)(25,64,35,74)(26,73,36,63)(27,62,37,72)(28,71,38,61)(29,80,39,70)(30,69,40,79)(41,134,51,124)(42,123,52,133)(43,132,53,122)(44,121,54,131)(45,130,55,140)(46,139,56,129)(47,128,57,138)(48,137,58,127)(49,126,59,136)(50,135,60,125)(101,151,111,141)(102,160,112,150)(103,149,113,159)(104,158,114,148)(105,147,115,157)(106,156,116,146)(107,145,117,155)(108,154,118,144)(109,143,119,153)(110,152,120,142)>;
G:=Group( (1,106,11,116)(2,71,12,61)(3,108,13,118)(4,73,14,63)(5,110,15,120)(6,75,16,65)(7,112,17,102)(8,77,18,67)(9,114,19,104)(10,79,20,69)(21,86,31,96)(22,131,32,121)(23,88,33,98)(24,133,34,123)(25,90,35,100)(26,135,36,125)(27,92,37,82)(28,137,38,127)(29,94,39,84)(30,139,40,129)(41,64,51,74)(42,101,52,111)(43,66,53,76)(44,103,54,113)(45,68,55,78)(46,105,56,115)(47,70,57,80)(48,107,58,117)(49,72,59,62)(50,109,60,119)(81,153,91,143)(83,155,93,145)(85,157,95,147)(87,159,97,149)(89,141,99,151)(122,150,132,160)(124,152,134,142)(126,154,136,144)(128,156,138,146)(130,158,140,148), (1,35,47,142)(2,26,48,153)(3,37,49,144)(4,28,50,155)(5,39,51,146)(6,30,52,157)(7,21,53,148)(8,32,54,159)(9,23,55,150)(10,34,56,141)(11,25,57,152)(12,36,58,143)(13,27,59,154)(14,38,60,145)(15,29,41,156)(16,40,42,147)(17,31,43,158)(18,22,44,149)(19,33,45,160)(20,24,46,151)(61,125,117,81)(62,136,118,92)(63,127,119,83)(64,138,120,94)(65,129,101,85)(66,140,102,96)(67,131,103,87)(68,122,104,98)(69,133,105,89)(70,124,106,100)(71,135,107,91)(72,126,108,82)(73,137,109,93)(74,128,110,84)(75,139,111,95)(76,130,112,86)(77,121,113,97)(78,132,114,88)(79,123,115,99)(80,134,116,90), (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), (1,84,11,94)(2,93,12,83)(3,82,13,92)(4,91,14,81)(5,100,15,90)(6,89,16,99)(7,98,17,88)(8,87,18,97)(9,96,19,86)(10,85,20,95)(21,68,31,78)(22,77,32,67)(23,66,33,76)(24,75,34,65)(25,64,35,74)(26,73,36,63)(27,62,37,72)(28,71,38,61)(29,80,39,70)(30,69,40,79)(41,134,51,124)(42,123,52,133)(43,132,53,122)(44,121,54,131)(45,130,55,140)(46,139,56,129)(47,128,57,138)(48,137,58,127)(49,126,59,136)(50,135,60,125)(101,151,111,141)(102,160,112,150)(103,149,113,159)(104,158,114,148)(105,147,115,157)(106,156,116,146)(107,145,117,155)(108,154,118,144)(109,143,119,153)(110,152,120,142) );
G=PermutationGroup([[(1,106,11,116),(2,71,12,61),(3,108,13,118),(4,73,14,63),(5,110,15,120),(6,75,16,65),(7,112,17,102),(8,77,18,67),(9,114,19,104),(10,79,20,69),(21,86,31,96),(22,131,32,121),(23,88,33,98),(24,133,34,123),(25,90,35,100),(26,135,36,125),(27,92,37,82),(28,137,38,127),(29,94,39,84),(30,139,40,129),(41,64,51,74),(42,101,52,111),(43,66,53,76),(44,103,54,113),(45,68,55,78),(46,105,56,115),(47,70,57,80),(48,107,58,117),(49,72,59,62),(50,109,60,119),(81,153,91,143),(83,155,93,145),(85,157,95,147),(87,159,97,149),(89,141,99,151),(122,150,132,160),(124,152,134,142),(126,154,136,144),(128,156,138,146),(130,158,140,148)], [(1,35,47,142),(2,26,48,153),(3,37,49,144),(4,28,50,155),(5,39,51,146),(6,30,52,157),(7,21,53,148),(8,32,54,159),(9,23,55,150),(10,34,56,141),(11,25,57,152),(12,36,58,143),(13,27,59,154),(14,38,60,145),(15,29,41,156),(16,40,42,147),(17,31,43,158),(18,22,44,149),(19,33,45,160),(20,24,46,151),(61,125,117,81),(62,136,118,92),(63,127,119,83),(64,138,120,94),(65,129,101,85),(66,140,102,96),(67,131,103,87),(68,122,104,98),(69,133,105,89),(70,124,106,100),(71,135,107,91),(72,126,108,82),(73,137,109,93),(74,128,110,84),(75,139,111,95),(76,130,112,86),(77,121,113,97),(78,132,114,88),(79,123,115,99),(80,134,116,90)], [(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)], [(1,84,11,94),(2,93,12,83),(3,82,13,92),(4,91,14,81),(5,100,15,90),(6,89,16,99),(7,98,17,88),(8,87,18,97),(9,96,19,86),(10,85,20,95),(21,68,31,78),(22,77,32,67),(23,66,33,76),(24,75,34,65),(25,64,35,74),(26,73,36,63),(27,62,37,72),(28,71,38,61),(29,80,39,70),(30,69,40,79),(41,134,51,124),(42,123,52,133),(43,132,53,122),(44,121,54,131),(45,130,55,140),(46,139,56,129),(47,128,57,138),(48,137,58,127),(49,126,59,136),(50,135,60,125),(101,151,111,141),(102,160,112,150),(103,149,113,159),(104,158,114,148),(105,147,115,157),(106,156,116,146),(107,145,117,155),(108,154,118,144),(109,143,119,153),(110,152,120,142)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | 2- 1+4 | Q8.10D10 | D5×C4○D4 |
| kernel | C42.151D10 | C42⋊D5 | Dic5⋊3Q8 | Dic5.Q8 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | Dic5 | D10 | C42 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 12 | 1 | 4 | 8 |
Matrix representation of C42.151D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 34 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,32,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,7,0,0,0,0,34,34,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
C42.151D10 in GAP, Magma, Sage, TeX
C_4^2._{151}D_{10} % in TeX
G:=Group("C4^2.151D10"); // GroupNames label
G:=SmallGroup(320,1365);
// by ID
G=gap.SmallGroup(320,1365);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,346,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations