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⋊D20.15C2, 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
Generators and relations for C42.131D10
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 >
Subgroups: 814 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C23.36C23, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8×C10, D5×C42, C42⋊D5, C4×D20, C4×D20, C4.Dic10, D10.13D4, C4⋊D20, C4⋊C4⋊D5, D10⋊3Q8, C20.23D4, Q8×C20, C42.131D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23.36C23, C4○D20, Q8⋊2D5, C23×D5, C2×C4○D20, C2×Q8⋊2D5, D5×C4○D4, C42.131D10
►On 160 points
Generators in S160
(1 68 58 113)(2 114 59 69)(3 70 60 115)(4 116 41 71)(5 72 42 117)(6 118 43 73)(7 74 44 119)(8 120 45 75)(9 76 46 101)(10 102 47 77)(11 78 48 103)(12 104 49 79)(13 80 50 105)(14 106 51 61)(15 62 52 107)(16 108 53 63)(17 64 54 109)(18 110 55 65)(19 66 56 111)(20 112 57 67)(21 143 140 90)(22 91 121 144)(23 145 122 92)(24 93 123 146)(25 147 124 94)(26 95 125 148)(27 149 126 96)(28 97 127 150)(29 151 128 98)(30 99 129 152)(31 153 130 100)(32 81 131 154)(33 155 132 82)(34 83 133 156)(35 157 134 84)(36 85 135 158)(37 159 136 86)(38 87 137 160)(39 141 138 88)(40 89 139 142)
(1 127 48 38)(2 128 49 39)(3 129 50 40)(4 130 51 21)(5 131 52 22)(6 132 53 23)(7 133 54 24)(8 134 55 25)(9 135 56 26)(10 136 57 27)(11 137 58 28)(12 138 59 29)(13 139 60 30)(14 140 41 31)(15 121 42 32)(16 122 43 33)(17 123 44 34)(18 124 45 35)(19 125 46 36)(20 126 47 37)(61 143 116 100)(62 144 117 81)(63 145 118 82)(64 146 119 83)(65 147 120 84)(66 148 101 85)(67 149 102 86)(68 150 103 87)(69 151 104 88)(70 152 105 89)(71 153 106 90)(72 154 107 91)(73 155 108 92)(74 156 109 93)(75 157 110 94)(76 158 111 95)(77 159 112 96)(78 160 113 97)(79 141 114 98)(80 142 115 99)
(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)
(1 107 11 117)(2 116 12 106)(3 105 13 115)(4 114 14 104)(5 103 15 113)(6 112 16 102)(7 101 17 111)(8 110 18 120)(9 119 19 109)(10 108 20 118)(21 141 31 151)(22 150 32 160)(23 159 33 149)(24 148 34 158)(25 157 35 147)(26 146 36 156)(27 155 37 145)(28 144 38 154)(29 153 39 143)(30 142 40 152)(41 69 51 79)(42 78 52 68)(43 67 53 77)(44 76 54 66)(45 65 55 75)(46 74 56 64)(47 63 57 73)(48 72 58 62)(49 61 59 71)(50 70 60 80)(81 127 91 137)(82 136 92 126)(83 125 93 135)(84 134 94 124)(85 123 95 133)(86 132 96 122)(87 121 97 131)(88 130 98 140)(89 139 99 129)(90 128 100 138)
G:=sub<Sym(160)| (1,68,58,113)(2,114,59,69)(3,70,60,115)(4,116,41,71)(5,72,42,117)(6,118,43,73)(7,74,44,119)(8,120,45,75)(9,76,46,101)(10,102,47,77)(11,78,48,103)(12,104,49,79)(13,80,50,105)(14,106,51,61)(15,62,52,107)(16,108,53,63)(17,64,54,109)(18,110,55,65)(19,66,56,111)(20,112,57,67)(21,143,140,90)(22,91,121,144)(23,145,122,92)(24,93,123,146)(25,147,124,94)(26,95,125,148)(27,149,126,96)(28,97,127,150)(29,151,128,98)(30,99,129,152)(31,153,130,100)(32,81,131,154)(33,155,132,82)(34,83,133,156)(35,157,134,84)(36,85,135,158)(37,159,136,86)(38,87,137,160)(39,141,138,88)(40,89,139,142), (1,127,48,38)(2,128,49,39)(3,129,50,40)(4,130,51,21)(5,131,52,22)(6,132,53,23)(7,133,54,24)(8,134,55,25)(9,135,56,26)(10,136,57,27)(11,137,58,28)(12,138,59,29)(13,139,60,30)(14,140,41,31)(15,121,42,32)(16,122,43,33)(17,123,44,34)(18,124,45,35)(19,125,46,36)(20,126,47,37)(61,143,116,100)(62,144,117,81)(63,145,118,82)(64,146,119,83)(65,147,120,84)(66,148,101,85)(67,149,102,86)(68,150,103,87)(69,151,104,88)(70,152,105,89)(71,153,106,90)(72,154,107,91)(73,155,108,92)(74,156,109,93)(75,157,110,94)(76,158,111,95)(77,159,112,96)(78,160,113,97)(79,141,114,98)(80,142,115,99), (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), (1,107,11,117)(2,116,12,106)(3,105,13,115)(4,114,14,104)(5,103,15,113)(6,112,16,102)(7,101,17,111)(8,110,18,120)(9,119,19,109)(10,108,20,118)(21,141,31,151)(22,150,32,160)(23,159,33,149)(24,148,34,158)(25,157,35,147)(26,146,36,156)(27,155,37,145)(28,144,38,154)(29,153,39,143)(30,142,40,152)(41,69,51,79)(42,78,52,68)(43,67,53,77)(44,76,54,66)(45,65,55,75)(46,74,56,64)(47,63,57,73)(48,72,58,62)(49,61,59,71)(50,70,60,80)(81,127,91,137)(82,136,92,126)(83,125,93,135)(84,134,94,124)(85,123,95,133)(86,132,96,122)(87,121,97,131)(88,130,98,140)(89,139,99,129)(90,128,100,138)>;
G:=Group( (1,68,58,113)(2,114,59,69)(3,70,60,115)(4,116,41,71)(5,72,42,117)(6,118,43,73)(7,74,44,119)(8,120,45,75)(9,76,46,101)(10,102,47,77)(11,78,48,103)(12,104,49,79)(13,80,50,105)(14,106,51,61)(15,62,52,107)(16,108,53,63)(17,64,54,109)(18,110,55,65)(19,66,56,111)(20,112,57,67)(21,143,140,90)(22,91,121,144)(23,145,122,92)(24,93,123,146)(25,147,124,94)(26,95,125,148)(27,149,126,96)(28,97,127,150)(29,151,128,98)(30,99,129,152)(31,153,130,100)(32,81,131,154)(33,155,132,82)(34,83,133,156)(35,157,134,84)(36,85,135,158)(37,159,136,86)(38,87,137,160)(39,141,138,88)(40,89,139,142), (1,127,48,38)(2,128,49,39)(3,129,50,40)(4,130,51,21)(5,131,52,22)(6,132,53,23)(7,133,54,24)(8,134,55,25)(9,135,56,26)(10,136,57,27)(11,137,58,28)(12,138,59,29)(13,139,60,30)(14,140,41,31)(15,121,42,32)(16,122,43,33)(17,123,44,34)(18,124,45,35)(19,125,46,36)(20,126,47,37)(61,143,116,100)(62,144,117,81)(63,145,118,82)(64,146,119,83)(65,147,120,84)(66,148,101,85)(67,149,102,86)(68,150,103,87)(69,151,104,88)(70,152,105,89)(71,153,106,90)(72,154,107,91)(73,155,108,92)(74,156,109,93)(75,157,110,94)(76,158,111,95)(77,159,112,96)(78,160,113,97)(79,141,114,98)(80,142,115,99), (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), (1,107,11,117)(2,116,12,106)(3,105,13,115)(4,114,14,104)(5,103,15,113)(6,112,16,102)(7,101,17,111)(8,110,18,120)(9,119,19,109)(10,108,20,118)(21,141,31,151)(22,150,32,160)(23,159,33,149)(24,148,34,158)(25,157,35,147)(26,146,36,156)(27,155,37,145)(28,144,38,154)(29,153,39,143)(30,142,40,152)(41,69,51,79)(42,78,52,68)(43,67,53,77)(44,76,54,66)(45,65,55,75)(46,74,56,64)(47,63,57,73)(48,72,58,62)(49,61,59,71)(50,70,60,80)(81,127,91,137)(82,136,92,126)(83,125,93,135)(84,134,94,124)(85,123,95,133)(86,132,96,122)(87,121,97,131)(88,130,98,140)(89,139,99,129)(90,128,100,138) );
G=PermutationGroup([[(1,68,58,113),(2,114,59,69),(3,70,60,115),(4,116,41,71),(5,72,42,117),(6,118,43,73),(7,74,44,119),(8,120,45,75),(9,76,46,101),(10,102,47,77),(11,78,48,103),(12,104,49,79),(13,80,50,105),(14,106,51,61),(15,62,52,107),(16,108,53,63),(17,64,54,109),(18,110,55,65),(19,66,56,111),(20,112,57,67),(21,143,140,90),(22,91,121,144),(23,145,122,92),(24,93,123,146),(25,147,124,94),(26,95,125,148),(27,149,126,96),(28,97,127,150),(29,151,128,98),(30,99,129,152),(31,153,130,100),(32,81,131,154),(33,155,132,82),(34,83,133,156),(35,157,134,84),(36,85,135,158),(37,159,136,86),(38,87,137,160),(39,141,138,88),(40,89,139,142)], [(1,127,48,38),(2,128,49,39),(3,129,50,40),(4,130,51,21),(5,131,52,22),(6,132,53,23),(7,133,54,24),(8,134,55,25),(9,135,56,26),(10,136,57,27),(11,137,58,28),(12,138,59,29),(13,139,60,30),(14,140,41,31),(15,121,42,32),(16,122,43,33),(17,123,44,34),(18,124,45,35),(19,125,46,36),(20,126,47,37),(61,143,116,100),(62,144,117,81),(63,145,118,82),(64,146,119,83),(65,147,120,84),(66,148,101,85),(67,149,102,86),(68,150,103,87),(69,151,104,88),(70,152,105,89),(71,153,106,90),(72,154,107,91),(73,155,108,92),(74,156,109,93),(75,157,110,94),(76,158,111,95),(77,159,112,96),(78,160,113,97),(79,141,114,98),(80,142,115,99)], [(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)], [(1,107,11,117),(2,116,12,106),(3,105,13,115),(4,114,14,104),(5,103,15,113),(6,112,16,102),(7,101,17,111),(8,110,18,120),(9,119,19,109),(10,108,20,118),(21,141,31,151),(22,150,32,160),(23,159,33,149),(24,148,34,158),(25,157,35,147),(26,146,36,156),(27,155,37,145),(28,144,38,154),(29,153,39,143),(30,142,40,152),(41,69,51,79),(42,78,52,68),(43,67,53,77),(44,76,54,66),(45,65,55,75),(46,74,56,64),(47,63,57,73),(48,72,58,62),(49,61,59,71),(50,70,60,80),(81,127,91,137),(82,136,92,126),(83,125,93,135),(84,134,94,124),(85,123,95,133),(86,132,96,122),(87,121,97,131),(88,130,98,140),(89,139,99,129),(90,128,100,138)]])
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⋊D20 | 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 |
Matrix representation of C42.131D10 ►in 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] >;
C42.131D10 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