metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.152D10, C10.962- (1+4), C42.C2⋊8D5, C4⋊C4.208D10, (C4×D20).25C2, D10⋊2Q8⋊38C2, (C4×Dic10)⋊48C2, (C2×C20).90C23, C4.Dic10⋊35C2, D10.39(C4○D4), C20.129(C4○D4), (C2×C10).238C24, (C4×C20).197C22, C4.38(Q8⋊2D5), D10.13D4.3C2, (C2×D20).233C22, C4⋊Dic5.243C22, C22.259(C23×D5), (C2×Dic5).123C23, (C4×Dic5).152C22, (C22×D5).103C23, C2.58(D4.10D10), D10⋊C4.138C22, C5⋊10(C22.46C24), (C2×Dic10).307C22, C10.D4.123C22, (D5×C4⋊C4)⋊38C2, C2.89(D5×C4○D4), C4⋊C4⋊D5⋊36C2, C4⋊C4⋊7D5⋊37C2, C10.200(C2×C4○D4), C2.23(C2×Q8⋊2D5), (C5×C42.C2)⋊11C2, (C2×C4×D5).137C22, (C2×C4).81(C22×D5), (C5×C4⋊C4).193C22, SmallGroup(320,1366)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 710 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×5], C2×C10, C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C42.C2 [×2], C42⋊2C2 [×2], Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], C22.46C24, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, C4×Dic10, C4×D20, C4.Dic10 [×2], D5×C4⋊C4, C4⋊C4⋊7D5, C4⋊C4⋊7D5 [×2], D10.13D4 [×2], D10⋊2Q8 [×2], C4⋊C4⋊D5 [×2], C5×C42.C2, C42.152D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D5 [×7], C22.46C24, Q8⋊2D5 [×2], C23×D5, C2×Q8⋊2D5, D5×C4○D4, D4.10D10, C42.152D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 118 25 53)(2 54 26 119)(3 120 27 55)(4 56 28 101)(5 102 29 57)(6 58 30 103)(7 104 31 59)(8 60 32 105)(9 106 33 41)(10 42 34 107)(11 108 35 43)(12 44 36 109)(13 110 37 45)(14 46 38 111)(15 112 39 47)(16 48 40 113)(17 114 21 49)(18 50 22 115)(19 116 23 51)(20 52 24 117)(61 141 134 90)(62 91 135 142)(63 143 136 92)(64 93 137 144)(65 145 138 94)(66 95 139 146)(67 147 140 96)(68 97 121 148)(69 149 122 98)(70 99 123 150)(71 151 124 100)(72 81 125 152)(73 153 126 82)(74 83 127 154)(75 155 128 84)(76 85 129 156)(77 157 130 86)(78 87 131 158)(79 159 132 88)(80 89 133 160)
(1 153 11 143)(2 93 12 83)(3 155 13 145)(4 95 14 85)(5 157 15 147)(6 97 16 87)(7 159 17 149)(8 99 18 89)(9 141 19 151)(10 81 20 91)(21 98 31 88)(22 160 32 150)(23 100 33 90)(24 142 34 152)(25 82 35 92)(26 144 36 154)(27 84 37 94)(28 146 38 156)(29 86 39 96)(30 148 40 158)(41 61 51 71)(42 125 52 135)(43 63 53 73)(44 127 54 137)(45 65 55 75)(46 129 56 139)(47 67 57 77)(48 131 58 121)(49 69 59 79)(50 133 60 123)(62 107 72 117)(64 109 74 119)(66 111 76 101)(68 113 78 103)(70 115 80 105)(102 130 112 140)(104 132 114 122)(106 134 116 124)(108 136 118 126)(110 138 120 128)
(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)
(1 152 11 142)(2 141 12 151)(3 150 13 160)(4 159 14 149)(5 148 15 158)(6 157 16 147)(7 146 17 156)(8 155 18 145)(9 144 19 154)(10 153 20 143)(21 85 31 95)(22 94 32 84)(23 83 33 93)(24 92 34 82)(25 81 35 91)(26 90 36 100)(27 99 37 89)(28 88 38 98)(29 97 39 87)(30 86 40 96)(41 64 51 74)(42 73 52 63)(43 62 53 72)(44 71 54 61)(45 80 55 70)(46 69 56 79)(47 78 57 68)(48 67 58 77)(49 76 59 66)(50 65 60 75)(101 132 111 122)(102 121 112 131)(103 130 113 140)(104 139 114 129)(105 128 115 138)(106 137 116 127)(107 126 117 136)(108 135 118 125)(109 124 119 134)(110 133 120 123)
G:=sub<Sym(160)| (1,118,25,53)(2,54,26,119)(3,120,27,55)(4,56,28,101)(5,102,29,57)(6,58,30,103)(7,104,31,59)(8,60,32,105)(9,106,33,41)(10,42,34,107)(11,108,35,43)(12,44,36,109)(13,110,37,45)(14,46,38,111)(15,112,39,47)(16,48,40,113)(17,114,21,49)(18,50,22,115)(19,116,23,51)(20,52,24,117)(61,141,134,90)(62,91,135,142)(63,143,136,92)(64,93,137,144)(65,145,138,94)(66,95,139,146)(67,147,140,96)(68,97,121,148)(69,149,122,98)(70,99,123,150)(71,151,124,100)(72,81,125,152)(73,153,126,82)(74,83,127,154)(75,155,128,84)(76,85,129,156)(77,157,130,86)(78,87,131,158)(79,159,132,88)(80,89,133,160), (1,153,11,143)(2,93,12,83)(3,155,13,145)(4,95,14,85)(5,157,15,147)(6,97,16,87)(7,159,17,149)(8,99,18,89)(9,141,19,151)(10,81,20,91)(21,98,31,88)(22,160,32,150)(23,100,33,90)(24,142,34,152)(25,82,35,92)(26,144,36,154)(27,84,37,94)(28,146,38,156)(29,86,39,96)(30,148,40,158)(41,61,51,71)(42,125,52,135)(43,63,53,73)(44,127,54,137)(45,65,55,75)(46,129,56,139)(47,67,57,77)(48,131,58,121)(49,69,59,79)(50,133,60,123)(62,107,72,117)(64,109,74,119)(66,111,76,101)(68,113,78,103)(70,115,80,105)(102,130,112,140)(104,132,114,122)(106,134,116,124)(108,136,118,126)(110,138,120,128), (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), (1,152,11,142)(2,141,12,151)(3,150,13,160)(4,159,14,149)(5,148,15,158)(6,157,16,147)(7,146,17,156)(8,155,18,145)(9,144,19,154)(10,153,20,143)(21,85,31,95)(22,94,32,84)(23,83,33,93)(24,92,34,82)(25,81,35,91)(26,90,36,100)(27,99,37,89)(28,88,38,98)(29,97,39,87)(30,86,40,96)(41,64,51,74)(42,73,52,63)(43,62,53,72)(44,71,54,61)(45,80,55,70)(46,69,56,79)(47,78,57,68)(48,67,58,77)(49,76,59,66)(50,65,60,75)(101,132,111,122)(102,121,112,131)(103,130,113,140)(104,139,114,129)(105,128,115,138)(106,137,116,127)(107,126,117,136)(108,135,118,125)(109,124,119,134)(110,133,120,123)>;
G:=Group( (1,118,25,53)(2,54,26,119)(3,120,27,55)(4,56,28,101)(5,102,29,57)(6,58,30,103)(7,104,31,59)(8,60,32,105)(9,106,33,41)(10,42,34,107)(11,108,35,43)(12,44,36,109)(13,110,37,45)(14,46,38,111)(15,112,39,47)(16,48,40,113)(17,114,21,49)(18,50,22,115)(19,116,23,51)(20,52,24,117)(61,141,134,90)(62,91,135,142)(63,143,136,92)(64,93,137,144)(65,145,138,94)(66,95,139,146)(67,147,140,96)(68,97,121,148)(69,149,122,98)(70,99,123,150)(71,151,124,100)(72,81,125,152)(73,153,126,82)(74,83,127,154)(75,155,128,84)(76,85,129,156)(77,157,130,86)(78,87,131,158)(79,159,132,88)(80,89,133,160), (1,153,11,143)(2,93,12,83)(3,155,13,145)(4,95,14,85)(5,157,15,147)(6,97,16,87)(7,159,17,149)(8,99,18,89)(9,141,19,151)(10,81,20,91)(21,98,31,88)(22,160,32,150)(23,100,33,90)(24,142,34,152)(25,82,35,92)(26,144,36,154)(27,84,37,94)(28,146,38,156)(29,86,39,96)(30,148,40,158)(41,61,51,71)(42,125,52,135)(43,63,53,73)(44,127,54,137)(45,65,55,75)(46,129,56,139)(47,67,57,77)(48,131,58,121)(49,69,59,79)(50,133,60,123)(62,107,72,117)(64,109,74,119)(66,111,76,101)(68,113,78,103)(70,115,80,105)(102,130,112,140)(104,132,114,122)(106,134,116,124)(108,136,118,126)(110,138,120,128), (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), (1,152,11,142)(2,141,12,151)(3,150,13,160)(4,159,14,149)(5,148,15,158)(6,157,16,147)(7,146,17,156)(8,155,18,145)(9,144,19,154)(10,153,20,143)(21,85,31,95)(22,94,32,84)(23,83,33,93)(24,92,34,82)(25,81,35,91)(26,90,36,100)(27,99,37,89)(28,88,38,98)(29,97,39,87)(30,86,40,96)(41,64,51,74)(42,73,52,63)(43,62,53,72)(44,71,54,61)(45,80,55,70)(46,69,56,79)(47,78,57,68)(48,67,58,77)(49,76,59,66)(50,65,60,75)(101,132,111,122)(102,121,112,131)(103,130,113,140)(104,139,114,129)(105,128,115,138)(106,137,116,127)(107,126,117,136)(108,135,118,125)(109,124,119,134)(110,133,120,123) );
G=PermutationGroup([(1,118,25,53),(2,54,26,119),(3,120,27,55),(4,56,28,101),(5,102,29,57),(6,58,30,103),(7,104,31,59),(8,60,32,105),(9,106,33,41),(10,42,34,107),(11,108,35,43),(12,44,36,109),(13,110,37,45),(14,46,38,111),(15,112,39,47),(16,48,40,113),(17,114,21,49),(18,50,22,115),(19,116,23,51),(20,52,24,117),(61,141,134,90),(62,91,135,142),(63,143,136,92),(64,93,137,144),(65,145,138,94),(66,95,139,146),(67,147,140,96),(68,97,121,148),(69,149,122,98),(70,99,123,150),(71,151,124,100),(72,81,125,152),(73,153,126,82),(74,83,127,154),(75,155,128,84),(76,85,129,156),(77,157,130,86),(78,87,131,158),(79,159,132,88),(80,89,133,160)], [(1,153,11,143),(2,93,12,83),(3,155,13,145),(4,95,14,85),(5,157,15,147),(6,97,16,87),(7,159,17,149),(8,99,18,89),(9,141,19,151),(10,81,20,91),(21,98,31,88),(22,160,32,150),(23,100,33,90),(24,142,34,152),(25,82,35,92),(26,144,36,154),(27,84,37,94),(28,146,38,156),(29,86,39,96),(30,148,40,158),(41,61,51,71),(42,125,52,135),(43,63,53,73),(44,127,54,137),(45,65,55,75),(46,129,56,139),(47,67,57,77),(48,131,58,121),(49,69,59,79),(50,133,60,123),(62,107,72,117),(64,109,74,119),(66,111,76,101),(68,113,78,103),(70,115,80,105),(102,130,112,140),(104,132,114,122),(106,134,116,124),(108,136,118,126),(110,138,120,128)], [(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)], [(1,152,11,142),(2,141,12,151),(3,150,13,160),(4,159,14,149),(5,148,15,158),(6,157,16,147),(7,146,17,156),(8,155,18,145),(9,144,19,154),(10,153,20,143),(21,85,31,95),(22,94,32,84),(23,83,33,93),(24,92,34,82),(25,81,35,91),(26,90,36,100),(27,99,37,89),(28,88,38,98),(29,97,39,87),(30,86,40,96),(41,64,51,74),(42,73,52,63),(43,62,53,72),(44,71,54,61),(45,80,55,70),(46,69,56,79),(47,78,57,68),(48,67,58,77),(49,76,59,66),(50,65,60,75),(101,132,111,122),(102,121,112,131),(103,130,113,140),(104,139,114,129),(105,128,115,138),(106,137,116,127),(107,126,117,136),(108,135,118,125),(109,124,119,134),(110,133,120,123)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 9 | 23 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 0 | 40 | 9 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 35 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 1 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,9,0,0,0,0,0,1],[9,0,0,0,0,0,23,32,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,32,40,0,0,0,0,39,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,7,8,0,0,0,0,35,34,0,0,0,0,0,0,9,1,0,0,0,0,2,32] >;
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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | 2- (1+4) | Q8⋊2D5 | D5×C4○D4 | D4.10D10 |
| kernel | C42.152D10 | C4×Dic10 | C4×D20 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C20 | D10 | C42 | C4⋊C4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{152}D_{10} % in TeX
G:=Group("C4^2.152D10"); // GroupNames label
G:=SmallGroup(320,1366);
// by ID
G=gap.SmallGroup(320,1366);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^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=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀