metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.188D10, C4○D20⋊16C4, D20⋊34(C2×C4), C4⋊C4.309D10, (D5×C42)⋊17C2, D20⋊8C4⋊45C2, Dic10⋊32(C2×C4), C42⋊D5⋊29C2, C42⋊C2⋊20D5, C10.39(C23×C4), (C2×C10).66C24, Dic5⋊10(C4○D4), Dic5⋊3Q8⋊44C2, Dic5⋊4D4⋊49C2, C20.202(C22×C4), (C4×C20).232C22, (C2×C20).877C23, C22⋊C4.126D10, D10.14(C22×C4), (C22×C4).364D10, C22.28(C23×D5), (C2×D20).264C22, Dic5.15(C22×C4), C23.154(C22×D5), (C22×C20).226C22, (C22×C10).136C23, (C4×Dic5).281C22, (C2×Dic5).373C23, (C22×D5).173C23, D10⋊C4.118C22, (C2×Dic10).292C22, C10.D4.131C22, (C22×Dic5).239C22, C5⋊3(C4×C4○D4), (C2×C4)⋊10(C4×D5), (C2×C4×Dic5)⋊6C2, C2.2(D5×C4○D4), C4.118(C2×C4×D5), (C2×C20)⋊25(C2×C4), (C4×D5)⋊14(C2×C4), C5⋊D4⋊10(C2×C4), C22.6(C2×C4×D5), C2.20(D5×C22×C4), (C5×C42⋊C2)⋊8C2, (C2×C4○D20).17C2, C10.132(C2×C4○D4), (C2×C4×D5).372C22, (C5×C4⋊C4).305C22, (C2×C4).272(C22×D5), (C2×C10).123(C22×C4), (C2×C5⋊D4).105C22, (C5×C22⋊C4).136C22, SmallGroup(320,1194)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.188D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >
Subgroups: 926 in 310 conjugacy classes, 155 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4×C4○D4, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, D5×C42, C42⋊D5, Dic5⋊4D4, Dic5⋊3Q8, D20⋊8C4, C2×C4×Dic5, C5×C42⋊C2, C2×C4○D20, C42.188D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, C24, D10, C23×C4, C2×C4○D4, C4×D5, C22×D5, C4×C4○D4, C2×C4×D5, C23×D5, D5×C22×C4, D5×C4○D4, C42.188D10
►On 160 points
Generators in S160
(1 143 63 92)(2 144 64 93)(3 145 65 94)(4 146 66 95)(5 147 67 96)(6 148 68 97)(7 149 69 98)(8 150 70 99)(9 141 61 100)(10 142 62 91)(11 45 89 129)(12 46 90 130)(13 47 81 121)(14 48 82 122)(15 49 83 123)(16 50 84 124)(17 41 85 125)(18 42 86 126)(19 43 87 127)(20 44 88 128)(21 152 72 101)(22 153 73 102)(23 154 74 103)(24 155 75 104)(25 156 76 105)(26 157 77 106)(27 158 78 107)(28 159 79 108)(29 160 80 109)(30 151 71 110)(31 136 56 111)(32 137 57 112)(33 138 58 113)(34 139 59 114)(35 140 60 115)(36 131 51 116)(37 132 52 117)(38 133 53 118)(39 134 54 119)(40 135 55 120)
(1 12 55 22)(2 81 56 74)(3 14 57 24)(4 83 58 76)(5 16 59 26)(6 85 60 78)(7 18 51 28)(8 87 52 80)(9 20 53 30)(10 89 54 72)(11 39 21 62)(13 31 23 64)(15 33 25 66)(17 35 27 68)(19 37 29 70)(32 75 65 82)(34 77 67 84)(36 79 69 86)(38 71 61 88)(40 73 63 90)(41 140 158 97)(42 116 159 149)(43 132 160 99)(44 118 151 141)(45 134 152 91)(46 120 153 143)(47 136 154 93)(48 112 155 145)(49 138 156 95)(50 114 157 147)(92 130 135 102)(94 122 137 104)(96 124 139 106)(98 126 131 108)(100 128 133 110)(101 142 129 119)(103 144 121 111)(105 146 123 113)(107 148 125 115)(109 150 127 117)
(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)
(1 139 63 114)(2 113 64 138)(3 137 65 112)(4 111 66 136)(5 135 67 120)(6 119 68 134)(7 133 69 118)(8 117 70 132)(9 131 61 116)(10 115 62 140)(11 107 89 158)(12 157 90 106)(13 105 81 156)(14 155 82 104)(15 103 83 154)(16 153 84 102)(17 101 85 152)(18 151 86 110)(19 109 87 160)(20 159 88 108)(21 125 72 41)(22 50 73 124)(23 123 74 49)(24 48 75 122)(25 121 76 47)(26 46 77 130)(27 129 78 45)(28 44 79 128)(29 127 80 43)(30 42 71 126)(31 95 56 146)(32 145 57 94)(33 93 58 144)(34 143 59 92)(35 91 60 142)(36 141 51 100)(37 99 52 150)(38 149 53 98)(39 97 54 148)(40 147 55 96)
G:=sub<Sym(160)| (1,143,63,92)(2,144,64,93)(3,145,65,94)(4,146,66,95)(5,147,67,96)(6,148,68,97)(7,149,69,98)(8,150,70,99)(9,141,61,100)(10,142,62,91)(11,45,89,129)(12,46,90,130)(13,47,81,121)(14,48,82,122)(15,49,83,123)(16,50,84,124)(17,41,85,125)(18,42,86,126)(19,43,87,127)(20,44,88,128)(21,152,72,101)(22,153,73,102)(23,154,74,103)(24,155,75,104)(25,156,76,105)(26,157,77,106)(27,158,78,107)(28,159,79,108)(29,160,80,109)(30,151,71,110)(31,136,56,111)(32,137,57,112)(33,138,58,113)(34,139,59,114)(35,140,60,115)(36,131,51,116)(37,132,52,117)(38,133,53,118)(39,134,54,119)(40,135,55,120), (1,12,55,22)(2,81,56,74)(3,14,57,24)(4,83,58,76)(5,16,59,26)(6,85,60,78)(7,18,51,28)(8,87,52,80)(9,20,53,30)(10,89,54,72)(11,39,21,62)(13,31,23,64)(15,33,25,66)(17,35,27,68)(19,37,29,70)(32,75,65,82)(34,77,67,84)(36,79,69,86)(38,71,61,88)(40,73,63,90)(41,140,158,97)(42,116,159,149)(43,132,160,99)(44,118,151,141)(45,134,152,91)(46,120,153,143)(47,136,154,93)(48,112,155,145)(49,138,156,95)(50,114,157,147)(92,130,135,102)(94,122,137,104)(96,124,139,106)(98,126,131,108)(100,128,133,110)(101,142,129,119)(103,144,121,111)(105,146,123,113)(107,148,125,115)(109,150,127,117), (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), (1,139,63,114)(2,113,64,138)(3,137,65,112)(4,111,66,136)(5,135,67,120)(6,119,68,134)(7,133,69,118)(8,117,70,132)(9,131,61,116)(10,115,62,140)(11,107,89,158)(12,157,90,106)(13,105,81,156)(14,155,82,104)(15,103,83,154)(16,153,84,102)(17,101,85,152)(18,151,86,110)(19,109,87,160)(20,159,88,108)(21,125,72,41)(22,50,73,124)(23,123,74,49)(24,48,75,122)(25,121,76,47)(26,46,77,130)(27,129,78,45)(28,44,79,128)(29,127,80,43)(30,42,71,126)(31,95,56,146)(32,145,57,94)(33,93,58,144)(34,143,59,92)(35,91,60,142)(36,141,51,100)(37,99,52,150)(38,149,53,98)(39,97,54,148)(40,147,55,96)>;
G:=Group( (1,143,63,92)(2,144,64,93)(3,145,65,94)(4,146,66,95)(5,147,67,96)(6,148,68,97)(7,149,69,98)(8,150,70,99)(9,141,61,100)(10,142,62,91)(11,45,89,129)(12,46,90,130)(13,47,81,121)(14,48,82,122)(15,49,83,123)(16,50,84,124)(17,41,85,125)(18,42,86,126)(19,43,87,127)(20,44,88,128)(21,152,72,101)(22,153,73,102)(23,154,74,103)(24,155,75,104)(25,156,76,105)(26,157,77,106)(27,158,78,107)(28,159,79,108)(29,160,80,109)(30,151,71,110)(31,136,56,111)(32,137,57,112)(33,138,58,113)(34,139,59,114)(35,140,60,115)(36,131,51,116)(37,132,52,117)(38,133,53,118)(39,134,54,119)(40,135,55,120), (1,12,55,22)(2,81,56,74)(3,14,57,24)(4,83,58,76)(5,16,59,26)(6,85,60,78)(7,18,51,28)(8,87,52,80)(9,20,53,30)(10,89,54,72)(11,39,21,62)(13,31,23,64)(15,33,25,66)(17,35,27,68)(19,37,29,70)(32,75,65,82)(34,77,67,84)(36,79,69,86)(38,71,61,88)(40,73,63,90)(41,140,158,97)(42,116,159,149)(43,132,160,99)(44,118,151,141)(45,134,152,91)(46,120,153,143)(47,136,154,93)(48,112,155,145)(49,138,156,95)(50,114,157,147)(92,130,135,102)(94,122,137,104)(96,124,139,106)(98,126,131,108)(100,128,133,110)(101,142,129,119)(103,144,121,111)(105,146,123,113)(107,148,125,115)(109,150,127,117), (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), (1,139,63,114)(2,113,64,138)(3,137,65,112)(4,111,66,136)(5,135,67,120)(6,119,68,134)(7,133,69,118)(8,117,70,132)(9,131,61,116)(10,115,62,140)(11,107,89,158)(12,157,90,106)(13,105,81,156)(14,155,82,104)(15,103,83,154)(16,153,84,102)(17,101,85,152)(18,151,86,110)(19,109,87,160)(20,159,88,108)(21,125,72,41)(22,50,73,124)(23,123,74,49)(24,48,75,122)(25,121,76,47)(26,46,77,130)(27,129,78,45)(28,44,79,128)(29,127,80,43)(30,42,71,126)(31,95,56,146)(32,145,57,94)(33,93,58,144)(34,143,59,92)(35,91,60,142)(36,141,51,100)(37,99,52,150)(38,149,53,98)(39,97,54,148)(40,147,55,96) );
G=PermutationGroup([[(1,143,63,92),(2,144,64,93),(3,145,65,94),(4,146,66,95),(5,147,67,96),(6,148,68,97),(7,149,69,98),(8,150,70,99),(9,141,61,100),(10,142,62,91),(11,45,89,129),(12,46,90,130),(13,47,81,121),(14,48,82,122),(15,49,83,123),(16,50,84,124),(17,41,85,125),(18,42,86,126),(19,43,87,127),(20,44,88,128),(21,152,72,101),(22,153,73,102),(23,154,74,103),(24,155,75,104),(25,156,76,105),(26,157,77,106),(27,158,78,107),(28,159,79,108),(29,160,80,109),(30,151,71,110),(31,136,56,111),(32,137,57,112),(33,138,58,113),(34,139,59,114),(35,140,60,115),(36,131,51,116),(37,132,52,117),(38,133,53,118),(39,134,54,119),(40,135,55,120)], [(1,12,55,22),(2,81,56,74),(3,14,57,24),(4,83,58,76),(5,16,59,26),(6,85,60,78),(7,18,51,28),(8,87,52,80),(9,20,53,30),(10,89,54,72),(11,39,21,62),(13,31,23,64),(15,33,25,66),(17,35,27,68),(19,37,29,70),(32,75,65,82),(34,77,67,84),(36,79,69,86),(38,71,61,88),(40,73,63,90),(41,140,158,97),(42,116,159,149),(43,132,160,99),(44,118,151,141),(45,134,152,91),(46,120,153,143),(47,136,154,93),(48,112,155,145),(49,138,156,95),(50,114,157,147),(92,130,135,102),(94,122,137,104),(96,124,139,106),(98,126,131,108),(100,128,133,110),(101,142,129,119),(103,144,121,111),(105,146,123,113),(107,148,125,115),(109,150,127,117)], [(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)], [(1,139,63,114),(2,113,64,138),(3,137,65,112),(4,111,66,136),(5,135,67,120),(6,119,68,134),(7,133,69,118),(8,117,70,132),(9,131,61,116),(10,115,62,140),(11,107,89,158),(12,157,90,106),(13,105,81,156),(14,155,82,104),(15,103,83,154),(16,153,84,102),(17,101,85,152),(18,151,86,110),(19,109,87,160),(20,159,88,108),(21,125,72,41),(22,50,73,124),(23,123,74,49),(24,48,75,122),(25,121,76,47),(26,46,77,130),(27,129,78,45),(28,44,79,128),(29,127,80,43),(30,42,71,126),(31,95,56,146),(32,145,57,94),(33,93,58,144),(34,143,59,92),(35,91,60,142),(36,141,51,100),(37,99,52,150),(38,149,53,98),(39,97,54,148),(40,147,55,96)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 4W | ··· | 4AD | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | C4×D5 | D5×C4○D4 |
| kernel | C42.188D10 | D5×C42 | C42⋊D5 | Dic5⋊4D4 | Dic5⋊3Q8 | D20⋊8C4 | C2×C4×Dic5 | C5×C42⋊C2 | C2×C4○D20 | C4○D20 | C42⋊C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 8 |
Matrix representation of C42.188D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 40 |
| 6 | 6 | 0 | 0 |
| 35 | 1 | 0 | 0 |
| 0 | 0 | 9 | 25 |
| 0 | 0 | 5 | 32 |
| 6 | 6 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 5 | 32 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,21,40],[6,35,0,0,6,1,0,0,0,0,9,5,0,0,25,32],[6,1,0,0,6,35,0,0,0,0,9,5,0,0,0,32] >;
C42.188D10 in GAP, Magma, Sage, TeX
C_4^2._{188}D_{10} % in TeX
G:=Group("C4^2.188D10"); // GroupNames label
G:=SmallGroup(320,1194);
// by ID
G=gap.SmallGroup(320,1194);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations