metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.131D10, (C4×Q8)⋊13D5, (C4×D20)⋊40C2, (Q8×C20)⋊15C2, (D5×C42)⋊7C2, C4⋊C4.298D10, D10⋊3Q8⋊46C2, D10.9(C4○D4), C4.48(C4○D20), C4⋊2D20.15C2, C42⋊D5⋊17C2, (C2×Q8).179D10, C4.Dic10⋊47C2, C20.340(C4○D4), C20.23D4⋊33C2, (C4×C20).176C22, (C2×C10).124C24, (C2×C20).622C23, C4.60(Q8⋊2D5), D10.13D4⋊50C2, (C2×D20).225C22, C4⋊Dic5.308C22, (Q8×C10).224C22, C22.145(C23×D5), C5⋊5(C23.36C23), (C2×Dic5).227C23, (C4×Dic5).337C22, (C22×D5).191C23, D10⋊C4.104C22, C10.D4.156C22, C2.31(D5×C4○D4), C4⋊C4⋊D5⋊51C2, C2.63(C2×C4○D20), C10.146(C2×C4○D4), C2.12(C2×Q8⋊2D5), (C2×C4×D5).379C22, (C5×C4⋊C4).352C22, (C2×C4).170(C22×D5), SmallGroup(320,1252)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 814 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×3], D5 [×4], C10 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×5], C20 [×4], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C23.36C23, C4×Dic5 [×3], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5 [×3], C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q8×C10, D5×C42, C42⋊D5 [×2], C4×D20, C4×D20 [×2], C4.Dic10, D10.13D4 [×2], C4⋊2D20, C4⋊C4⋊D5 [×2], D10⋊3Q8, C20.23D4, Q8×C20, C42.131D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C4○D20 [×2], Q8⋊2D5 [×2], C23×D5, C2×C4○D20, C2×Q8⋊2D5, D5×C4○D4, C42.131D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=c9 >
►On 160 points
Generators in S160
(1 27 54 68)(2 69 55 28)(3 29 56 70)(4 71 57 30)(5 31 58 72)(6 73 59 32)(7 33 60 74)(8 75 41 34)(9 35 42 76)(10 77 43 36)(11 37 44 78)(12 79 45 38)(13 39 46 80)(14 61 47 40)(15 21 48 62)(16 63 49 22)(17 23 50 64)(18 65 51 24)(19 25 52 66)(20 67 53 26)(81 151 105 132)(82 133 106 152)(83 153 107 134)(84 135 108 154)(85 155 109 136)(86 137 110 156)(87 157 111 138)(88 139 112 158)(89 159 113 140)(90 121 114 160)(91 141 115 122)(92 123 116 142)(93 143 117 124)(94 125 118 144)(95 145 119 126)(96 127 120 146)(97 147 101 128)(98 129 102 148)(99 149 103 130)(100 131 104 150)
(1 147 44 138)(2 148 45 139)(3 149 46 140)(4 150 47 121)(5 151 48 122)(6 152 49 123)(7 153 50 124)(8 154 51 125)(9 155 52 126)(10 156 53 127)(11 157 54 128)(12 158 55 129)(13 159 56 130)(14 160 57 131)(15 141 58 132)(16 142 59 133)(17 143 60 134)(18 144 41 135)(19 145 42 136)(20 146 43 137)(21 115 72 81)(22 116 73 82)(23 117 74 83)(24 118 75 84)(25 119 76 85)(26 120 77 86)(27 101 78 87)(28 102 79 88)(29 103 80 89)(30 104 61 90)(31 105 62 91)(32 106 63 92)(33 107 64 93)(34 108 65 94)(35 109 66 95)(36 110 67 96)(37 111 68 97)(38 112 69 98)(39 113 70 99)(40 114 71 100)
(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 62 11 72)(2 71 12 61)(3 80 13 70)(4 69 14 79)(5 78 15 68)(6 67 16 77)(7 76 17 66)(8 65 18 75)(9 74 19 64)(10 63 20 73)(21 44 31 54)(22 53 32 43)(23 42 33 52)(24 51 34 41)(25 60 35 50)(26 49 36 59)(27 58 37 48)(28 47 38 57)(29 56 39 46)(30 45 40 55)(81 147 91 157)(82 156 92 146)(83 145 93 155)(84 154 94 144)(85 143 95 153)(86 152 96 142)(87 141 97 151)(88 150 98 160)(89 159 99 149)(90 148 100 158)(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)| (1,27,54,68)(2,69,55,28)(3,29,56,70)(4,71,57,30)(5,31,58,72)(6,73,59,32)(7,33,60,74)(8,75,41,34)(9,35,42,76)(10,77,43,36)(11,37,44,78)(12,79,45,38)(13,39,46,80)(14,61,47,40)(15,21,48,62)(16,63,49,22)(17,23,50,64)(18,65,51,24)(19,25,52,66)(20,67,53,26)(81,151,105,132)(82,133,106,152)(83,153,107,134)(84,135,108,154)(85,155,109,136)(86,137,110,156)(87,157,111,138)(88,139,112,158)(89,159,113,140)(90,121,114,160)(91,141,115,122)(92,123,116,142)(93,143,117,124)(94,125,118,144)(95,145,119,126)(96,127,120,146)(97,147,101,128)(98,129,102,148)(99,149,103,130)(100,131,104,150), (1,147,44,138)(2,148,45,139)(3,149,46,140)(4,150,47,121)(5,151,48,122)(6,152,49,123)(7,153,50,124)(8,154,51,125)(9,155,52,126)(10,156,53,127)(11,157,54,128)(12,158,55,129)(13,159,56,130)(14,160,57,131)(15,141,58,132)(16,142,59,133)(17,143,60,134)(18,144,41,135)(19,145,42,136)(20,146,43,137)(21,115,72,81)(22,116,73,82)(23,117,74,83)(24,118,75,84)(25,119,76,85)(26,120,77,86)(27,101,78,87)(28,102,79,88)(29,103,80,89)(30,104,61,90)(31,105,62,91)(32,106,63,92)(33,107,64,93)(34,108,65,94)(35,109,66,95)(36,110,67,96)(37,111,68,97)(38,112,69,98)(39,113,70,99)(40,114,71,100), (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), (1,62,11,72)(2,71,12,61)(3,80,13,70)(4,69,14,79)(5,78,15,68)(6,67,16,77)(7,76,17,66)(8,65,18,75)(9,74,19,64)(10,63,20,73)(21,44,31,54)(22,53,32,43)(23,42,33,52)(24,51,34,41)(25,60,35,50)(26,49,36,59)(27,58,37,48)(28,47,38,57)(29,56,39,46)(30,45,40,55)(81,147,91,157)(82,156,92,146)(83,145,93,155)(84,154,94,144)(85,143,95,153)(86,152,96,142)(87,141,97,151)(88,150,98,160)(89,159,99,149)(90,148,100,158)(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( (1,27,54,68)(2,69,55,28)(3,29,56,70)(4,71,57,30)(5,31,58,72)(6,73,59,32)(7,33,60,74)(8,75,41,34)(9,35,42,76)(10,77,43,36)(11,37,44,78)(12,79,45,38)(13,39,46,80)(14,61,47,40)(15,21,48,62)(16,63,49,22)(17,23,50,64)(18,65,51,24)(19,25,52,66)(20,67,53,26)(81,151,105,132)(82,133,106,152)(83,153,107,134)(84,135,108,154)(85,155,109,136)(86,137,110,156)(87,157,111,138)(88,139,112,158)(89,159,113,140)(90,121,114,160)(91,141,115,122)(92,123,116,142)(93,143,117,124)(94,125,118,144)(95,145,119,126)(96,127,120,146)(97,147,101,128)(98,129,102,148)(99,149,103,130)(100,131,104,150), (1,147,44,138)(2,148,45,139)(3,149,46,140)(4,150,47,121)(5,151,48,122)(6,152,49,123)(7,153,50,124)(8,154,51,125)(9,155,52,126)(10,156,53,127)(11,157,54,128)(12,158,55,129)(13,159,56,130)(14,160,57,131)(15,141,58,132)(16,142,59,133)(17,143,60,134)(18,144,41,135)(19,145,42,136)(20,146,43,137)(21,115,72,81)(22,116,73,82)(23,117,74,83)(24,118,75,84)(25,119,76,85)(26,120,77,86)(27,101,78,87)(28,102,79,88)(29,103,80,89)(30,104,61,90)(31,105,62,91)(32,106,63,92)(33,107,64,93)(34,108,65,94)(35,109,66,95)(36,110,67,96)(37,111,68,97)(38,112,69,98)(39,113,70,99)(40,114,71,100), (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), (1,62,11,72)(2,71,12,61)(3,80,13,70)(4,69,14,79)(5,78,15,68)(6,67,16,77)(7,76,17,66)(8,65,18,75)(9,74,19,64)(10,63,20,73)(21,44,31,54)(22,53,32,43)(23,42,33,52)(24,51,34,41)(25,60,35,50)(26,49,36,59)(27,58,37,48)(28,47,38,57)(29,56,39,46)(30,45,40,55)(81,147,91,157)(82,156,92,146)(83,145,93,155)(84,154,94,144)(85,143,95,153)(86,152,96,142)(87,141,97,151)(88,150,98,160)(89,159,99,149)(90,148,100,158)(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([(1,27,54,68),(2,69,55,28),(3,29,56,70),(4,71,57,30),(5,31,58,72),(6,73,59,32),(7,33,60,74),(8,75,41,34),(9,35,42,76),(10,77,43,36),(11,37,44,78),(12,79,45,38),(13,39,46,80),(14,61,47,40),(15,21,48,62),(16,63,49,22),(17,23,50,64),(18,65,51,24),(19,25,52,66),(20,67,53,26),(81,151,105,132),(82,133,106,152),(83,153,107,134),(84,135,108,154),(85,155,109,136),(86,137,110,156),(87,157,111,138),(88,139,112,158),(89,159,113,140),(90,121,114,160),(91,141,115,122),(92,123,116,142),(93,143,117,124),(94,125,118,144),(95,145,119,126),(96,127,120,146),(97,147,101,128),(98,129,102,148),(99,149,103,130),(100,131,104,150)], [(1,147,44,138),(2,148,45,139),(3,149,46,140),(4,150,47,121),(5,151,48,122),(6,152,49,123),(7,153,50,124),(8,154,51,125),(9,155,52,126),(10,156,53,127),(11,157,54,128),(12,158,55,129),(13,159,56,130),(14,160,57,131),(15,141,58,132),(16,142,59,133),(17,143,60,134),(18,144,41,135),(19,145,42,136),(20,146,43,137),(21,115,72,81),(22,116,73,82),(23,117,74,83),(24,118,75,84),(25,119,76,85),(26,120,77,86),(27,101,78,87),(28,102,79,88),(29,103,80,89),(30,104,61,90),(31,105,62,91),(32,106,63,92),(33,107,64,93),(34,108,65,94),(35,109,66,95),(36,110,67,96),(37,111,68,97),(38,112,69,98),(39,113,70,99),(40,114,71,100)], [(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)], [(1,62,11,72),(2,71,12,61),(3,80,13,70),(4,69,14,79),(5,78,15,68),(6,67,16,77),(7,76,17,66),(8,65,18,75),(9,74,19,64),(10,63,20,73),(21,44,31,54),(22,53,32,43),(23,42,33,52),(24,51,34,41),(25,60,35,50),(26,49,36,59),(27,58,37,48),(28,47,38,57),(29,56,39,46),(30,45,40,55),(81,147,91,157),(82,156,92,146),(83,145,93,155),(84,154,94,144),(85,143,95,153),(86,152,96,142),(87,141,97,151),(88,150,98,160),(89,159,99,149),(90,148,100,158),(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 ⊆ GL4(𝔽41) generated by
| 25 | 15 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 18 | 6 |
| 0 | 0 | 35 | 23 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 25 | 16 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 9 | 13 |
| 0 | 0 | 28 | 13 |
| 16 | 25 | 0 | 0 |
| 39 | 25 | 0 | 0 |
| 0 | 0 | 23 | 35 |
| 0 | 0 | 20 | 18 |
G:=sub<GL(4,GF(41))| [25,2,0,0,15,16,0,0,0,0,18,35,0,0,6,23],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[25,2,0,0,16,16,0,0,0,0,9,28,0,0,13,13],[16,39,0,0,25,25,0,0,0,0,23,20,0,0,35,18] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | C4○D20 | Q8⋊2D5 | D5×C4○D4 |
| kernel | C42.131D10 | D5×C42 | C42⋊D5 | C4×D20 | C4.Dic10 | D10.13D4 | C4⋊2D20 | C4⋊C4⋊D5 | D10⋊3Q8 | C20.23D4 | Q8×C20 | C4×Q8 | C20 | D10 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 6 | 6 | 2 | 16 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{131}D_{10} % in TeX
G:=Group("C4^2.131D10"); // GroupNames label
G:=SmallGroup(320,1252);
// by ID
G=gap.SmallGroup(320,1252);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,794,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀