metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.155D10, C10.972- 1+4, C4⋊C4.211D10, C20⋊2Q8⋊33C2, C42.C2⋊11D5, (C2×C20).92C23, C4.Dic10⋊37C2, C42⋊D5.7C2, Dic5⋊3Q8⋊38C2, C20.131(C4○D4), (C4×C20).200C22, (C2×C10).241C24, D10⋊2Q8.13C2, C4.20(Q8⋊2D5), C4⋊Dic5.244C22, C22.262(C23×D5), C5⋊5(C22.35C24), (C4×Dic5).154C22, (C2×Dic5).271C23, (C22×D5).106C23, C2.60(D4.10D10), D10⋊C4.112C22, (C2×Dic10).188C22, C10.D4.124C22, C4⋊C4⋊D5.3C2, C10.118(C2×C4○D4), (C5×C42.C2)⋊14C2, C2.25(C2×Q8⋊2D5), (C2×C4×D5).140C22, (C5×C4⋊C4).196C22, (C2×C4).206(C22×D5), SmallGroup(320,1369)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.155D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2b2c9 >
Subgroups: 590 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22.35C24, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C20⋊2Q8, C42⋊D5, Dic5⋊3Q8, C4.Dic10, D10⋊2Q8, C4⋊C4⋊D5, C5×C42.C2, C42.155D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.35C24, Q8⋊2D5, C23×D5, C2×Q8⋊2D5, D4.10D10, C42.155D10
►On 160 points
Generators in S160
(1 51 100 109)(2 110 81 52)(3 53 82 111)(4 112 83 54)(5 55 84 113)(6 114 85 56)(7 57 86 115)(8 116 87 58)(9 59 88 117)(10 118 89 60)(11 41 90 119)(12 120 91 42)(13 43 92 101)(14 102 93 44)(15 45 94 103)(16 104 95 46)(17 47 96 105)(18 106 97 48)(19 49 98 107)(20 108 99 50)(21 159 70 135)(22 136 71 160)(23 141 72 137)(24 138 73 142)(25 143 74 139)(26 140 75 144)(27 145 76 121)(28 122 77 146)(29 147 78 123)(30 124 79 148)(31 149 80 125)(32 126 61 150)(33 151 62 127)(34 128 63 152)(35 153 64 129)(36 130 65 154)(37 155 66 131)(38 132 67 156)(39 157 68 133)(40 134 69 158)
(1 158 11 148)(2 125 12 135)(3 160 13 150)(4 127 14 137)(5 142 15 152)(6 129 16 139)(7 144 17 154)(8 131 18 121)(9 146 19 156)(10 133 20 123)(21 110 31 120)(22 43 32 53)(23 112 33 102)(24 45 34 55)(25 114 35 104)(26 47 36 57)(27 116 37 106)(28 49 38 59)(29 118 39 108)(30 51 40 41)(42 70 52 80)(44 72 54 62)(46 74 56 64)(48 76 58 66)(50 78 60 68)(61 111 71 101)(63 113 73 103)(65 115 75 105)(67 117 77 107)(69 119 79 109)(81 149 91 159)(82 136 92 126)(83 151 93 141)(84 138 94 128)(85 153 95 143)(86 140 96 130)(87 155 97 145)(88 122 98 132)(89 157 99 147)(90 124 100 134)
(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 10 100 89)(2 88 81 9)(3 8 82 87)(4 86 83 7)(5 6 84 85)(11 20 90 99)(12 98 91 19)(13 18 92 97)(14 96 93 17)(15 16 94 95)(21 28 70 77)(22 76 71 27)(23 26 72 75)(24 74 73 25)(29 40 78 69)(30 68 79 39)(31 38 80 67)(32 66 61 37)(33 36 62 65)(34 64 63 35)(41 50 119 108)(42 107 120 49)(43 48 101 106)(44 105 102 47)(45 46 103 104)(51 60 109 118)(52 117 110 59)(53 58 111 116)(54 115 112 57)(55 56 113 114)(121 136 145 160)(122 159 146 135)(123 134 147 158)(124 157 148 133)(125 132 149 156)(126 155 150 131)(127 130 151 154)(128 153 152 129)(137 140 141 144)(138 143 142 139)
G:=sub<Sym(160)| (1,51,100,109)(2,110,81,52)(3,53,82,111)(4,112,83,54)(5,55,84,113)(6,114,85,56)(7,57,86,115)(8,116,87,58)(9,59,88,117)(10,118,89,60)(11,41,90,119)(12,120,91,42)(13,43,92,101)(14,102,93,44)(15,45,94,103)(16,104,95,46)(17,47,96,105)(18,106,97,48)(19,49,98,107)(20,108,99,50)(21,159,70,135)(22,136,71,160)(23,141,72,137)(24,138,73,142)(25,143,74,139)(26,140,75,144)(27,145,76,121)(28,122,77,146)(29,147,78,123)(30,124,79,148)(31,149,80,125)(32,126,61,150)(33,151,62,127)(34,128,63,152)(35,153,64,129)(36,130,65,154)(37,155,66,131)(38,132,67,156)(39,157,68,133)(40,134,69,158), (1,158,11,148)(2,125,12,135)(3,160,13,150)(4,127,14,137)(5,142,15,152)(6,129,16,139)(7,144,17,154)(8,131,18,121)(9,146,19,156)(10,133,20,123)(21,110,31,120)(22,43,32,53)(23,112,33,102)(24,45,34,55)(25,114,35,104)(26,47,36,57)(27,116,37,106)(28,49,38,59)(29,118,39,108)(30,51,40,41)(42,70,52,80)(44,72,54,62)(46,74,56,64)(48,76,58,66)(50,78,60,68)(61,111,71,101)(63,113,73,103)(65,115,75,105)(67,117,77,107)(69,119,79,109)(81,149,91,159)(82,136,92,126)(83,151,93,141)(84,138,94,128)(85,153,95,143)(86,140,96,130)(87,155,97,145)(88,122,98,132)(89,157,99,147)(90,124,100,134), (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), (1,10,100,89)(2,88,81,9)(3,8,82,87)(4,86,83,7)(5,6,84,85)(11,20,90,99)(12,98,91,19)(13,18,92,97)(14,96,93,17)(15,16,94,95)(21,28,70,77)(22,76,71,27)(23,26,72,75)(24,74,73,25)(29,40,78,69)(30,68,79,39)(31,38,80,67)(32,66,61,37)(33,36,62,65)(34,64,63,35)(41,50,119,108)(42,107,120,49)(43,48,101,106)(44,105,102,47)(45,46,103,104)(51,60,109,118)(52,117,110,59)(53,58,111,116)(54,115,112,57)(55,56,113,114)(121,136,145,160)(122,159,146,135)(123,134,147,158)(124,157,148,133)(125,132,149,156)(126,155,150,131)(127,130,151,154)(128,153,152,129)(137,140,141,144)(138,143,142,139)>;
G:=Group( (1,51,100,109)(2,110,81,52)(3,53,82,111)(4,112,83,54)(5,55,84,113)(6,114,85,56)(7,57,86,115)(8,116,87,58)(9,59,88,117)(10,118,89,60)(11,41,90,119)(12,120,91,42)(13,43,92,101)(14,102,93,44)(15,45,94,103)(16,104,95,46)(17,47,96,105)(18,106,97,48)(19,49,98,107)(20,108,99,50)(21,159,70,135)(22,136,71,160)(23,141,72,137)(24,138,73,142)(25,143,74,139)(26,140,75,144)(27,145,76,121)(28,122,77,146)(29,147,78,123)(30,124,79,148)(31,149,80,125)(32,126,61,150)(33,151,62,127)(34,128,63,152)(35,153,64,129)(36,130,65,154)(37,155,66,131)(38,132,67,156)(39,157,68,133)(40,134,69,158), (1,158,11,148)(2,125,12,135)(3,160,13,150)(4,127,14,137)(5,142,15,152)(6,129,16,139)(7,144,17,154)(8,131,18,121)(9,146,19,156)(10,133,20,123)(21,110,31,120)(22,43,32,53)(23,112,33,102)(24,45,34,55)(25,114,35,104)(26,47,36,57)(27,116,37,106)(28,49,38,59)(29,118,39,108)(30,51,40,41)(42,70,52,80)(44,72,54,62)(46,74,56,64)(48,76,58,66)(50,78,60,68)(61,111,71,101)(63,113,73,103)(65,115,75,105)(67,117,77,107)(69,119,79,109)(81,149,91,159)(82,136,92,126)(83,151,93,141)(84,138,94,128)(85,153,95,143)(86,140,96,130)(87,155,97,145)(88,122,98,132)(89,157,99,147)(90,124,100,134), (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), (1,10,100,89)(2,88,81,9)(3,8,82,87)(4,86,83,7)(5,6,84,85)(11,20,90,99)(12,98,91,19)(13,18,92,97)(14,96,93,17)(15,16,94,95)(21,28,70,77)(22,76,71,27)(23,26,72,75)(24,74,73,25)(29,40,78,69)(30,68,79,39)(31,38,80,67)(32,66,61,37)(33,36,62,65)(34,64,63,35)(41,50,119,108)(42,107,120,49)(43,48,101,106)(44,105,102,47)(45,46,103,104)(51,60,109,118)(52,117,110,59)(53,58,111,116)(54,115,112,57)(55,56,113,114)(121,136,145,160)(122,159,146,135)(123,134,147,158)(124,157,148,133)(125,132,149,156)(126,155,150,131)(127,130,151,154)(128,153,152,129)(137,140,141,144)(138,143,142,139) );
G=PermutationGroup([[(1,51,100,109),(2,110,81,52),(3,53,82,111),(4,112,83,54),(5,55,84,113),(6,114,85,56),(7,57,86,115),(8,116,87,58),(9,59,88,117),(10,118,89,60),(11,41,90,119),(12,120,91,42),(13,43,92,101),(14,102,93,44),(15,45,94,103),(16,104,95,46),(17,47,96,105),(18,106,97,48),(19,49,98,107),(20,108,99,50),(21,159,70,135),(22,136,71,160),(23,141,72,137),(24,138,73,142),(25,143,74,139),(26,140,75,144),(27,145,76,121),(28,122,77,146),(29,147,78,123),(30,124,79,148),(31,149,80,125),(32,126,61,150),(33,151,62,127),(34,128,63,152),(35,153,64,129),(36,130,65,154),(37,155,66,131),(38,132,67,156),(39,157,68,133),(40,134,69,158)], [(1,158,11,148),(2,125,12,135),(3,160,13,150),(4,127,14,137),(5,142,15,152),(6,129,16,139),(7,144,17,154),(8,131,18,121),(9,146,19,156),(10,133,20,123),(21,110,31,120),(22,43,32,53),(23,112,33,102),(24,45,34,55),(25,114,35,104),(26,47,36,57),(27,116,37,106),(28,49,38,59),(29,118,39,108),(30,51,40,41),(42,70,52,80),(44,72,54,62),(46,74,56,64),(48,76,58,66),(50,78,60,68),(61,111,71,101),(63,113,73,103),(65,115,75,105),(67,117,77,107),(69,119,79,109),(81,149,91,159),(82,136,92,126),(83,151,93,141),(84,138,94,128),(85,153,95,143),(86,140,96,130),(87,155,97,145),(88,122,98,132),(89,157,99,147),(90,124,100,134)], [(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)], [(1,10,100,89),(2,88,81,9),(3,8,82,87),(4,86,83,7),(5,6,84,85),(11,20,90,99),(12,98,91,19),(13,18,92,97),(14,96,93,17),(15,16,94,95),(21,28,70,77),(22,76,71,27),(23,26,72,75),(24,74,73,25),(29,40,78,69),(30,68,79,39),(31,38,80,67),(32,66,61,37),(33,36,62,65),(34,64,63,35),(41,50,119,108),(42,107,120,49),(43,48,101,106),(44,105,102,47),(45,46,103,104),(51,60,109,118),(52,117,110,59),(53,58,111,116),(54,115,112,57),(55,56,113,114),(121,136,145,160),(122,159,146,135),(123,134,147,158),(124,157,148,133),(125,132,149,156),(126,155,150,131),(127,130,151,154),(128,153,152,129),(137,140,141,144),(138,143,142,139)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2- 1+4 | Q8⋊2D5 | D4.10D10 |
| kernel | C42.155D10 | C20⋊2Q8 | C42⋊D5 | Dic5⋊3Q8 | C4.Dic10 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C20 | C42 | C4⋊C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 4 | 2 | 12 | 2 | 4 | 8 |
Matrix representation of C42.155D10 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 30 |
| 0 | 0 | 0 | 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 22 | 19 | 32 | 9 | 0 | 0 | 0 | 0 |
| 22 | 12 | 32 | 10 | 0 | 0 | 0 | 0 |
| 32 | 9 | 19 | 22 | 0 | 0 | 0 | 0 |
| 32 | 10 | 19 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 8 | 21 | 18 |
| 0 | 0 | 0 | 0 | 18 | 25 | 20 | 0 |
| 0 | 0 | 0 | 0 | 20 | 23 | 34 | 33 |
| 0 | 0 | 0 | 0 | 21 | 0 | 23 | 16 |
| 19 | 22 | 9 | 32 | 0 | 0 | 0 | 0 |
| 12 | 22 | 10 | 32 | 0 | 0 | 0 | 0 |
| 9 | 32 | 22 | 19 | 0 | 0 | 0 | 0 |
| 10 | 32 | 29 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 | 21 | 3 |
| 0 | 0 | 0 | 0 | 23 | 7 | 20 | 20 |
| 0 | 0 | 0 | 0 | 21 | 3 | 34 | 1 |
| 0 | 0 | 0 | 0 | 20 | 20 | 23 | 7 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[22,22,32,32,0,0,0,0,19,12,9,10,0,0,0,0,32,32,19,19,0,0,0,0,9,10,22,29,0,0,0,0,0,0,0,0,7,18,20,21,0,0,0,0,8,25,23,0,0,0,0,0,21,20,34,23,0,0,0,0,18,0,33,16],[19,12,9,10,0,0,0,0,22,22,32,32,0,0,0,0,9,10,22,29,0,0,0,0,32,32,19,19,0,0,0,0,0,0,0,0,34,23,21,20,0,0,0,0,1,7,3,20,0,0,0,0,21,20,34,23,0,0,0,0,3,20,1,7] >;
C42.155D10 in GAP, Magma, Sage, TeX
C_4^2._{155}D_{10} % in TeX
G:=Group("C4^2.155D10"); // GroupNames label
G:=SmallGroup(320,1369);
// by ID
G=gap.SmallGroup(320,1369);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^9>;
// generators/relations