metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.132D10, C10.122- 1+4, (C4×Q8)⋊14D5, (Q8×C20)⋊16C2, C4⋊C4.299D10, D10⋊3Q8⋊9C2, (C4×D20).22C2, C4.68(C4○D20), C42⋊2D5⋊18C2, C42⋊D5⋊34C2, (C2×Q8).180D10, C20.6Q8⋊19C2, D10.32(C4○D4), C20.119(C4○D4), (C2×C10).125C24, (C4×C20).177C22, (C2×C20).623C23, Dic5.Q8⋊9C2, D10.13D4.1C2, (C2×D20).226C22, C4⋊Dic5.309C22, (Q8×C10).225C22, (C2×Dic5).56C23, (C22×D5).47C23, C22.146(C23×D5), D10⋊C4.89C22, C5⋊5(C22.46C24), (C4×Dic5).229C22, C10.D4.76C22, C2.13(Q8.10D10), (D5×C4⋊C4)⋊19C2, C2.32(D5×C4○D4), C4⋊C4⋊7D5⋊17C2, C2.64(C2×C4○D20), (C2×C4×D5).84C22, C10.147(C2×C4○D4), (C5×C4⋊C4).353C22, (C2×C4).289(C22×D5), SmallGroup(320,1253)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.132D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 694 in 214 conjugacy classes, 97 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×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.46C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, Q8×C10, C20.6Q8, C42⋊D5, C4×D20, C42⋊2D5, Dic5.Q8, D5×C4⋊C4, C4⋊C4⋊7D5, D10.13D4, D10⋊3Q8, Q8×C20, C42.132D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, C4○D20, C23×D5, C2×C4○D20, Q8.10D10, D5×C4○D4, C42.132D10
►On 160 points
Generators in S160
(1 148 73 57)(2 58 74 149)(3 150 75 59)(4 60 76 151)(5 152 77 41)(6 42 78 153)(7 154 79 43)(8 44 80 155)(9 156 61 45)(10 46 62 157)(11 158 63 47)(12 48 64 159)(13 160 65 49)(14 50 66 141)(15 142 67 51)(16 52 68 143)(17 144 69 53)(18 54 70 145)(19 146 71 55)(20 56 72 147)(21 96 134 119)(22 120 135 97)(23 98 136 101)(24 102 137 99)(25 100 138 103)(26 104 139 81)(27 82 140 105)(28 106 121 83)(29 84 122 107)(30 108 123 85)(31 86 124 109)(32 110 125 87)(33 88 126 111)(34 112 127 89)(35 90 128 113)(36 114 129 91)(37 92 130 115)(38 116 131 93)(39 94 132 117)(40 118 133 95)
(1 117 63 84)(2 118 64 85)(3 119 65 86)(4 120 66 87)(5 101 67 88)(6 102 68 89)(7 103 69 90)(8 104 70 91)(9 105 71 92)(10 106 72 93)(11 107 73 94)(12 108 74 95)(13 109 75 96)(14 110 76 97)(15 111 77 98)(16 112 78 99)(17 113 79 100)(18 114 80 81)(19 115 61 82)(20 116 62 83)(21 49 124 150)(22 50 125 151)(23 51 126 152)(24 52 127 153)(25 53 128 154)(26 54 129 155)(27 55 130 156)(28 56 131 157)(29 57 132 158)(30 58 133 159)(31 59 134 160)(32 60 135 141)(33 41 136 142)(34 42 137 143)(35 43 138 144)(36 44 139 145)(37 45 140 146)(38 46 121 147)(39 47 122 148)(40 48 123 149)
(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 38 11 28)(2 27 12 37)(3 36 13 26)(4 25 14 35)(5 34 15 24)(6 23 16 33)(7 32 17 22)(8 21 18 31)(9 30 19 40)(10 39 20 29)(41 112 51 102)(42 101 52 111)(43 110 53 120)(44 119 54 109)(45 108 55 118)(46 117 56 107)(47 106 57 116)(48 115 58 105)(49 104 59 114)(50 113 60 103)(61 123 71 133)(62 132 72 122)(63 121 73 131)(64 130 74 140)(65 139 75 129)(66 128 76 138)(67 137 77 127)(68 126 78 136)(69 135 79 125)(70 124 80 134)(81 150 91 160)(82 159 92 149)(83 148 93 158)(84 157 94 147)(85 146 95 156)(86 155 96 145)(87 144 97 154)(88 153 98 143)(89 142 99 152)(90 151 100 141)
G:=sub<Sym(160)| (1,148,73,57)(2,58,74,149)(3,150,75,59)(4,60,76,151)(5,152,77,41)(6,42,78,153)(7,154,79,43)(8,44,80,155)(9,156,61,45)(10,46,62,157)(11,158,63,47)(12,48,64,159)(13,160,65,49)(14,50,66,141)(15,142,67,51)(16,52,68,143)(17,144,69,53)(18,54,70,145)(19,146,71,55)(20,56,72,147)(21,96,134,119)(22,120,135,97)(23,98,136,101)(24,102,137,99)(25,100,138,103)(26,104,139,81)(27,82,140,105)(28,106,121,83)(29,84,122,107)(30,108,123,85)(31,86,124,109)(32,110,125,87)(33,88,126,111)(34,112,127,89)(35,90,128,113)(36,114,129,91)(37,92,130,115)(38,116,131,93)(39,94,132,117)(40,118,133,95), (1,117,63,84)(2,118,64,85)(3,119,65,86)(4,120,66,87)(5,101,67,88)(6,102,68,89)(7,103,69,90)(8,104,70,91)(9,105,71,92)(10,106,72,93)(11,107,73,94)(12,108,74,95)(13,109,75,96)(14,110,76,97)(15,111,77,98)(16,112,78,99)(17,113,79,100)(18,114,80,81)(19,115,61,82)(20,116,62,83)(21,49,124,150)(22,50,125,151)(23,51,126,152)(24,52,127,153)(25,53,128,154)(26,54,129,155)(27,55,130,156)(28,56,131,157)(29,57,132,158)(30,58,133,159)(31,59,134,160)(32,60,135,141)(33,41,136,142)(34,42,137,143)(35,43,138,144)(36,44,139,145)(37,45,140,146)(38,46,121,147)(39,47,122,148)(40,48,123,149), (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), (1,38,11,28)(2,27,12,37)(3,36,13,26)(4,25,14,35)(5,34,15,24)(6,23,16,33)(7,32,17,22)(8,21,18,31)(9,30,19,40)(10,39,20,29)(41,112,51,102)(42,101,52,111)(43,110,53,120)(44,119,54,109)(45,108,55,118)(46,117,56,107)(47,106,57,116)(48,115,58,105)(49,104,59,114)(50,113,60,103)(61,123,71,133)(62,132,72,122)(63,121,73,131)(64,130,74,140)(65,139,75,129)(66,128,76,138)(67,137,77,127)(68,126,78,136)(69,135,79,125)(70,124,80,134)(81,150,91,160)(82,159,92,149)(83,148,93,158)(84,157,94,147)(85,146,95,156)(86,155,96,145)(87,144,97,154)(88,153,98,143)(89,142,99,152)(90,151,100,141)>;
G:=Group( (1,148,73,57)(2,58,74,149)(3,150,75,59)(4,60,76,151)(5,152,77,41)(6,42,78,153)(7,154,79,43)(8,44,80,155)(9,156,61,45)(10,46,62,157)(11,158,63,47)(12,48,64,159)(13,160,65,49)(14,50,66,141)(15,142,67,51)(16,52,68,143)(17,144,69,53)(18,54,70,145)(19,146,71,55)(20,56,72,147)(21,96,134,119)(22,120,135,97)(23,98,136,101)(24,102,137,99)(25,100,138,103)(26,104,139,81)(27,82,140,105)(28,106,121,83)(29,84,122,107)(30,108,123,85)(31,86,124,109)(32,110,125,87)(33,88,126,111)(34,112,127,89)(35,90,128,113)(36,114,129,91)(37,92,130,115)(38,116,131,93)(39,94,132,117)(40,118,133,95), (1,117,63,84)(2,118,64,85)(3,119,65,86)(4,120,66,87)(5,101,67,88)(6,102,68,89)(7,103,69,90)(8,104,70,91)(9,105,71,92)(10,106,72,93)(11,107,73,94)(12,108,74,95)(13,109,75,96)(14,110,76,97)(15,111,77,98)(16,112,78,99)(17,113,79,100)(18,114,80,81)(19,115,61,82)(20,116,62,83)(21,49,124,150)(22,50,125,151)(23,51,126,152)(24,52,127,153)(25,53,128,154)(26,54,129,155)(27,55,130,156)(28,56,131,157)(29,57,132,158)(30,58,133,159)(31,59,134,160)(32,60,135,141)(33,41,136,142)(34,42,137,143)(35,43,138,144)(36,44,139,145)(37,45,140,146)(38,46,121,147)(39,47,122,148)(40,48,123,149), (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), (1,38,11,28)(2,27,12,37)(3,36,13,26)(4,25,14,35)(5,34,15,24)(6,23,16,33)(7,32,17,22)(8,21,18,31)(9,30,19,40)(10,39,20,29)(41,112,51,102)(42,101,52,111)(43,110,53,120)(44,119,54,109)(45,108,55,118)(46,117,56,107)(47,106,57,116)(48,115,58,105)(49,104,59,114)(50,113,60,103)(61,123,71,133)(62,132,72,122)(63,121,73,131)(64,130,74,140)(65,139,75,129)(66,128,76,138)(67,137,77,127)(68,126,78,136)(69,135,79,125)(70,124,80,134)(81,150,91,160)(82,159,92,149)(83,148,93,158)(84,157,94,147)(85,146,95,156)(86,155,96,145)(87,144,97,154)(88,153,98,143)(89,142,99,152)(90,151,100,141) );
G=PermutationGroup([[(1,148,73,57),(2,58,74,149),(3,150,75,59),(4,60,76,151),(5,152,77,41),(6,42,78,153),(7,154,79,43),(8,44,80,155),(9,156,61,45),(10,46,62,157),(11,158,63,47),(12,48,64,159),(13,160,65,49),(14,50,66,141),(15,142,67,51),(16,52,68,143),(17,144,69,53),(18,54,70,145),(19,146,71,55),(20,56,72,147),(21,96,134,119),(22,120,135,97),(23,98,136,101),(24,102,137,99),(25,100,138,103),(26,104,139,81),(27,82,140,105),(28,106,121,83),(29,84,122,107),(30,108,123,85),(31,86,124,109),(32,110,125,87),(33,88,126,111),(34,112,127,89),(35,90,128,113),(36,114,129,91),(37,92,130,115),(38,116,131,93),(39,94,132,117),(40,118,133,95)], [(1,117,63,84),(2,118,64,85),(3,119,65,86),(4,120,66,87),(5,101,67,88),(6,102,68,89),(7,103,69,90),(8,104,70,91),(9,105,71,92),(10,106,72,93),(11,107,73,94),(12,108,74,95),(13,109,75,96),(14,110,76,97),(15,111,77,98),(16,112,78,99),(17,113,79,100),(18,114,80,81),(19,115,61,82),(20,116,62,83),(21,49,124,150),(22,50,125,151),(23,51,126,152),(24,52,127,153),(25,53,128,154),(26,54,129,155),(27,55,130,156),(28,56,131,157),(29,57,132,158),(30,58,133,159),(31,59,134,160),(32,60,135,141),(33,41,136,142),(34,42,137,143),(35,43,138,144),(36,44,139,145),(37,45,140,146),(38,46,121,147),(39,47,122,148),(40,48,123,149)], [(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)], [(1,38,11,28),(2,27,12,37),(3,36,13,26),(4,25,14,35),(5,34,15,24),(6,23,16,33),(7,32,17,22),(8,21,18,31),(9,30,19,40),(10,39,20,29),(41,112,51,102),(42,101,52,111),(43,110,53,120),(44,119,54,109),(45,108,55,118),(46,117,56,107),(47,106,57,116),(48,115,58,105),(49,104,59,114),(50,113,60,103),(61,123,71,133),(62,132,72,122),(63,121,73,131),(64,130,74,140),(65,139,75,129),(66,128,76,138),(67,137,77,127),(68,126,78,136),(69,135,79,125),(70,124,80,134),(81,150,91,160),(82,159,92,149),(83,148,93,158),(84,157,94,147),(85,146,95,156),(86,155,96,145),(87,144,97,154),(88,153,98,143),(89,142,99,152),(90,151,100,141)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4R | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D10 | C4○D20 | 2- 1+4 | Q8.10D10 | D5×C4○D4 |
| kernel | C42.132D10 | C20.6Q8 | C42⋊D5 | C4×D20 | C42⋊2D5 | Dic5.Q8 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊3Q8 | Q8×C20 | C4×Q8 | C20 | D10 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.132D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 28 | 28 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 20 | 21 | 0 | 0 |
| 18 | 21 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,9,0,0,0,0,9],[28,13,0,0,28,32,0,0,0,0,0,1,0,0,1,0],[20,18,0,0,21,21,0,0,0,0,0,40,0,0,40,0] >;
C42.132D10 in GAP, Magma, Sage, TeX
C_4^2._{132}D_{10} % in TeX
G:=Group("C4^2.132D10"); // GroupNames label
G:=SmallGroup(320,1253);
// by ID
G=gap.SmallGroup(320,1253);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,136,12550]);
// 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=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations