metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.179D10, C10.852+ 1+4, C4⋊Q8⋊17D5, (C4×D20)⋊52C2, C4⋊D20⋊41C2, C4⋊C4.222D10, (C2×Q8).89D10, C20.141(C4○D4), C20.23D4⋊28C2, (C2×C10).278C24, (C4×C20).219C22, (C2×C20).640C23, C4.42(Q8⋊2D5), C2.89(D4⋊6D10), (C2×D20).283C22, C4⋊Dic5.387C22, (Q8×C10).145C22, C22.299(C23×D5), D10⋊C4.53C22, C5⋊5(C22.49C24), (C4×Dic5).175C22, (C2×Dic5).285C23, (C22×D5).123C23, (C5×C4⋊Q8)⋊20C2, C4⋊C4⋊7D5⋊44C2, C10.125(C2×C4○D4), C2.33(C2×Q8⋊2D5), (C2×C4×D5).160C22, (C5×C4⋊C4).221C22, (C2×C4).603(C22×D5), SmallGroup(320,1406)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.179D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=a2c9 >
Subgroups: 918 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.49C24, C4×Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C4×D20, C4⋊C4⋊7D5, C4⋊D20, C20.23D4, C5×C4⋊Q8, C42.179D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.49C24, Q8⋊2D5, C23×D5, D4⋊6D10, C2×Q8⋊2D5, C42.179D10
►On 160 points
Generators in S160
(1 82 149 70)(2 71 150 83)(3 84 151 72)(4 73 152 85)(5 86 153 74)(6 75 154 87)(7 88 155 76)(8 77 156 89)(9 90 157 78)(10 79 158 91)(11 92 159 80)(12 61 160 93)(13 94 141 62)(14 63 142 95)(15 96 143 64)(16 65 144 97)(17 98 145 66)(18 67 146 99)(19 100 147 68)(20 69 148 81)(21 57 119 121)(22 122 120 58)(23 59 101 123)(24 124 102 60)(25 41 103 125)(26 126 104 42)(27 43 105 127)(28 128 106 44)(29 45 107 129)(30 130 108 46)(31 47 109 131)(32 132 110 48)(33 49 111 133)(34 134 112 50)(35 51 113 135)(36 136 114 52)(37 53 115 137)(38 138 116 54)(39 55 117 139)(40 140 118 56)
(1 123 11 133)(2 134 12 124)(3 125 13 135)(4 136 14 126)(5 127 15 137)(6 138 16 128)(7 129 17 139)(8 140 18 130)(9 131 19 121)(10 122 20 132)(21 90 31 100)(22 81 32 91)(23 92 33 82)(24 83 34 93)(25 94 35 84)(26 85 36 95)(27 96 37 86)(28 87 38 97)(29 98 39 88)(30 89 40 99)(41 141 51 151)(42 152 52 142)(43 143 53 153)(44 154 54 144)(45 145 55 155)(46 156 56 146)(47 147 57 157)(48 158 58 148)(49 149 59 159)(50 160 60 150)(61 102 71 112)(62 113 72 103)(63 104 73 114)(64 115 74 105)(65 106 75 116)(66 117 76 107)(67 108 77 118)(68 119 78 109)(69 110 79 120)(70 101 80 111)
(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 27 159 115)(2 114 160 26)(3 25 141 113)(4 112 142 24)(5 23 143 111)(6 110 144 22)(7 21 145 109)(8 108 146 40)(9 39 147 107)(10 106 148 38)(11 37 149 105)(12 104 150 36)(13 35 151 103)(14 102 152 34)(15 33 153 101)(16 120 154 32)(17 31 155 119)(18 118 156 30)(19 29 157 117)(20 116 158 28)(41 94 135 72)(42 71 136 93)(43 92 137 70)(44 69 138 91)(45 90 139 68)(46 67 140 89)(47 88 121 66)(48 65 122 87)(49 86 123 64)(50 63 124 85)(51 84 125 62)(52 61 126 83)(53 82 127 80)(54 79 128 81)(55 100 129 78)(56 77 130 99)(57 98 131 76)(58 75 132 97)(59 96 133 74)(60 73 134 95)
G:=sub<Sym(160)| (1,82,149,70)(2,71,150,83)(3,84,151,72)(4,73,152,85)(5,86,153,74)(6,75,154,87)(7,88,155,76)(8,77,156,89)(9,90,157,78)(10,79,158,91)(11,92,159,80)(12,61,160,93)(13,94,141,62)(14,63,142,95)(15,96,143,64)(16,65,144,97)(17,98,145,66)(18,67,146,99)(19,100,147,68)(20,69,148,81)(21,57,119,121)(22,122,120,58)(23,59,101,123)(24,124,102,60)(25,41,103,125)(26,126,104,42)(27,43,105,127)(28,128,106,44)(29,45,107,129)(30,130,108,46)(31,47,109,131)(32,132,110,48)(33,49,111,133)(34,134,112,50)(35,51,113,135)(36,136,114,52)(37,53,115,137)(38,138,116,54)(39,55,117,139)(40,140,118,56), (1,123,11,133)(2,134,12,124)(3,125,13,135)(4,136,14,126)(5,127,15,137)(6,138,16,128)(7,129,17,139)(8,140,18,130)(9,131,19,121)(10,122,20,132)(21,90,31,100)(22,81,32,91)(23,92,33,82)(24,83,34,93)(25,94,35,84)(26,85,36,95)(27,96,37,86)(28,87,38,97)(29,98,39,88)(30,89,40,99)(41,141,51,151)(42,152,52,142)(43,143,53,153)(44,154,54,144)(45,145,55,155)(46,156,56,146)(47,147,57,157)(48,158,58,148)(49,149,59,159)(50,160,60,150)(61,102,71,112)(62,113,72,103)(63,104,73,114)(64,115,74,105)(65,106,75,116)(66,117,76,107)(67,108,77,118)(68,119,78,109)(69,110,79,120)(70,101,80,111), (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), (1,27,159,115)(2,114,160,26)(3,25,141,113)(4,112,142,24)(5,23,143,111)(6,110,144,22)(7,21,145,109)(8,108,146,40)(9,39,147,107)(10,106,148,38)(11,37,149,105)(12,104,150,36)(13,35,151,103)(14,102,152,34)(15,33,153,101)(16,120,154,32)(17,31,155,119)(18,118,156,30)(19,29,157,117)(20,116,158,28)(41,94,135,72)(42,71,136,93)(43,92,137,70)(44,69,138,91)(45,90,139,68)(46,67,140,89)(47,88,121,66)(48,65,122,87)(49,86,123,64)(50,63,124,85)(51,84,125,62)(52,61,126,83)(53,82,127,80)(54,79,128,81)(55,100,129,78)(56,77,130,99)(57,98,131,76)(58,75,132,97)(59,96,133,74)(60,73,134,95)>;
G:=Group( (1,82,149,70)(2,71,150,83)(3,84,151,72)(4,73,152,85)(5,86,153,74)(6,75,154,87)(7,88,155,76)(8,77,156,89)(9,90,157,78)(10,79,158,91)(11,92,159,80)(12,61,160,93)(13,94,141,62)(14,63,142,95)(15,96,143,64)(16,65,144,97)(17,98,145,66)(18,67,146,99)(19,100,147,68)(20,69,148,81)(21,57,119,121)(22,122,120,58)(23,59,101,123)(24,124,102,60)(25,41,103,125)(26,126,104,42)(27,43,105,127)(28,128,106,44)(29,45,107,129)(30,130,108,46)(31,47,109,131)(32,132,110,48)(33,49,111,133)(34,134,112,50)(35,51,113,135)(36,136,114,52)(37,53,115,137)(38,138,116,54)(39,55,117,139)(40,140,118,56), (1,123,11,133)(2,134,12,124)(3,125,13,135)(4,136,14,126)(5,127,15,137)(6,138,16,128)(7,129,17,139)(8,140,18,130)(9,131,19,121)(10,122,20,132)(21,90,31,100)(22,81,32,91)(23,92,33,82)(24,83,34,93)(25,94,35,84)(26,85,36,95)(27,96,37,86)(28,87,38,97)(29,98,39,88)(30,89,40,99)(41,141,51,151)(42,152,52,142)(43,143,53,153)(44,154,54,144)(45,145,55,155)(46,156,56,146)(47,147,57,157)(48,158,58,148)(49,149,59,159)(50,160,60,150)(61,102,71,112)(62,113,72,103)(63,104,73,114)(64,115,74,105)(65,106,75,116)(66,117,76,107)(67,108,77,118)(68,119,78,109)(69,110,79,120)(70,101,80,111), (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), (1,27,159,115)(2,114,160,26)(3,25,141,113)(4,112,142,24)(5,23,143,111)(6,110,144,22)(7,21,145,109)(8,108,146,40)(9,39,147,107)(10,106,148,38)(11,37,149,105)(12,104,150,36)(13,35,151,103)(14,102,152,34)(15,33,153,101)(16,120,154,32)(17,31,155,119)(18,118,156,30)(19,29,157,117)(20,116,158,28)(41,94,135,72)(42,71,136,93)(43,92,137,70)(44,69,138,91)(45,90,139,68)(46,67,140,89)(47,88,121,66)(48,65,122,87)(49,86,123,64)(50,63,124,85)(51,84,125,62)(52,61,126,83)(53,82,127,80)(54,79,128,81)(55,100,129,78)(56,77,130,99)(57,98,131,76)(58,75,132,97)(59,96,133,74)(60,73,134,95) );
G=PermutationGroup([[(1,82,149,70),(2,71,150,83),(3,84,151,72),(4,73,152,85),(5,86,153,74),(6,75,154,87),(7,88,155,76),(8,77,156,89),(9,90,157,78),(10,79,158,91),(11,92,159,80),(12,61,160,93),(13,94,141,62),(14,63,142,95),(15,96,143,64),(16,65,144,97),(17,98,145,66),(18,67,146,99),(19,100,147,68),(20,69,148,81),(21,57,119,121),(22,122,120,58),(23,59,101,123),(24,124,102,60),(25,41,103,125),(26,126,104,42),(27,43,105,127),(28,128,106,44),(29,45,107,129),(30,130,108,46),(31,47,109,131),(32,132,110,48),(33,49,111,133),(34,134,112,50),(35,51,113,135),(36,136,114,52),(37,53,115,137),(38,138,116,54),(39,55,117,139),(40,140,118,56)], [(1,123,11,133),(2,134,12,124),(3,125,13,135),(4,136,14,126),(5,127,15,137),(6,138,16,128),(7,129,17,139),(8,140,18,130),(9,131,19,121),(10,122,20,132),(21,90,31,100),(22,81,32,91),(23,92,33,82),(24,83,34,93),(25,94,35,84),(26,85,36,95),(27,96,37,86),(28,87,38,97),(29,98,39,88),(30,89,40,99),(41,141,51,151),(42,152,52,142),(43,143,53,153),(44,154,54,144),(45,145,55,155),(46,156,56,146),(47,147,57,157),(48,158,58,148),(49,149,59,159),(50,160,60,150),(61,102,71,112),(62,113,72,103),(63,104,73,114),(64,115,74,105),(65,106,75,116),(66,117,76,107),(67,108,77,118),(68,119,78,109),(69,110,79,120),(70,101,80,111)], [(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)], [(1,27,159,115),(2,114,160,26),(3,25,141,113),(4,112,142,24),(5,23,143,111),(6,110,144,22),(7,21,145,109),(8,108,146,40),(9,39,147,107),(10,106,148,38),(11,37,149,105),(12,104,150,36),(13,35,151,103),(14,102,152,34),(15,33,153,101),(16,120,154,32),(17,31,155,119),(18,118,156,30),(19,29,157,117),(20,116,158,28),(41,94,135,72),(42,71,136,93),(43,92,137,70),(44,69,138,91),(45,90,139,68),(46,67,140,89),(47,88,121,66),(48,65,122,87),(49,86,123,64),(50,63,124,85),(51,84,125,62),(52,61,126,83),(53,82,127,80),(54,79,128,81),(55,100,129,78),(56,77,130,99),(57,98,131,76),(58,75,132,97),(59,96,133,74),(60,73,134,95)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ 1+4 | Q8⋊2D5 | D4⋊6D10 |
| kernel | C42.179D10 | C4×D20 | C4⋊C4⋊7D5 | C4⋊D20 | C20.23D4 | C5×C4⋊Q8 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 8 | 2 | 8 | 4 | 1 | 8 | 4 |
Matrix representation of C42.179D10 ►in GL6(𝔽41)
| 13 | 34 | 0 | 0 | 0 | 0 |
| 36 | 28 | 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 | 0 | 40 |
| 28 | 7 | 0 | 0 | 0 | 0 |
| 5 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 20 | 9 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 4 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 37 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 38 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 6 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [13,36,0,0,0,0,34,28,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[28,5,0,0,0,0,7,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,20,0,0,0,0,0,9],[6,4,0,0,0,0,1,35,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,32,0,0,0,0,0,37,9],[1,0,0,0,0,0,38,40,0,0,0,0,0,0,34,33,0,0,0,0,6,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
C42.179D10 in GAP, Magma, Sage, TeX
C_4^2._{179}D_{10} % in TeX
G:=Group("C4^2.179D10"); // GroupNames label
G:=SmallGroup(320,1406);
// by ID
G=gap.SmallGroup(320,1406);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,570,185,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^9>;
// generators/relations