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, (C2×C10).66C24, C10.39(C23×C4), Dic5⋊10(C4○D4), Dic5⋊4D4⋊49C2, Dic5⋊3Q8⋊44C2, (C2×C20).877C23, (C4×C20).232C22, C20.202(C22×C4), 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×C10).136C23, (C22×C20).226C22, (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
Subgroups: 926 in 310 conjugacy classes, 155 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×14], C22, C22 [×2], C22 [×10], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C42 [×2], C42 [×8], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×8], Dic5 [×2], C20 [×4], C20 [×4], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42 [×3], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×C10, C4×C4○D4, C4×Dic5 [×2], C4×Dic5 [×6], C10.D4 [×4], D10⋊C4 [×4], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20, D5×C42 [×2], C42⋊D5 [×2], Dic5⋊4D4 [×4], Dic5⋊3Q8 [×2], D20⋊8C4 [×2], C2×C4×Dic5, C5×C42⋊C2, C2×C4○D20, C42.188D10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C4×C4○D4, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D5×C4○D4 [×2], C42.188D10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 143 63 131)(2 144 64 132)(3 145 65 133)(4 146 66 134)(5 147 67 135)(6 148 68 136)(7 149 69 137)(8 150 70 138)(9 141 61 139)(10 142 62 140)(11 44 88 91)(12 45 89 92)(13 46 90 93)(14 47 81 94)(15 48 82 95)(16 49 83 96)(17 50 84 97)(18 41 85 98)(19 42 86 99)(20 43 87 100)(21 113 33 101)(22 114 34 102)(23 115 35 103)(24 116 36 104)(25 117 37 105)(26 118 38 106)(27 119 39 107)(28 120 40 108)(29 111 31 109)(30 112 32 110)(51 158 78 129)(52 159 79 130)(53 160 80 121)(54 151 71 122)(55 152 72 123)(56 153 73 124)(57 154 74 125)(58 155 75 126)(59 156 76 127)(60 157 77 128)
(1 13 28 56)(2 81 29 74)(3 15 30 58)(4 83 21 76)(5 17 22 60)(6 85 23 78)(7 19 24 52)(8 87 25 80)(9 11 26 54)(10 89 27 72)(12 39 55 62)(14 31 57 64)(16 33 59 66)(18 35 51 68)(20 37 53 70)(32 75 65 82)(34 77 67 84)(36 79 69 86)(38 71 61 88)(40 73 63 90)(41 103 158 136)(42 116 159 149)(43 105 160 138)(44 118 151 141)(45 107 152 140)(46 120 153 143)(47 109 154 132)(48 112 155 145)(49 101 156 134)(50 114 157 147)(91 106 122 139)(92 119 123 142)(93 108 124 131)(94 111 125 144)(95 110 126 133)(96 113 127 146)(97 102 128 135)(98 115 129 148)(99 104 130 137)(100 117 121 150)
(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)
(1 102 63 114)(2 113 64 101)(3 110 65 112)(4 111 66 109)(5 108 67 120)(6 119 68 107)(7 106 69 118)(8 117 70 105)(9 104 61 116)(10 115 62 103)(11 159 88 130)(12 129 89 158)(13 157 90 128)(14 127 81 156)(15 155 82 126)(16 125 83 154)(17 153 84 124)(18 123 85 152)(19 151 86 122)(20 121 87 160)(21 144 33 132)(22 131 34 143)(23 142 35 140)(24 139 36 141)(25 150 37 138)(26 137 38 149)(27 148 39 136)(28 135 40 147)(29 146 31 134)(30 133 32 145)(41 55 98 72)(42 71 99 54)(43 53 100 80)(44 79 91 52)(45 51 92 78)(46 77 93 60)(47 59 94 76)(48 75 95 58)(49 57 96 74)(50 73 97 56)
G:=sub<Sym(160)| (1,143,63,131)(2,144,64,132)(3,145,65,133)(4,146,66,134)(5,147,67,135)(6,148,68,136)(7,149,69,137)(8,150,70,138)(9,141,61,139)(10,142,62,140)(11,44,88,91)(12,45,89,92)(13,46,90,93)(14,47,81,94)(15,48,82,95)(16,49,83,96)(17,50,84,97)(18,41,85,98)(19,42,86,99)(20,43,87,100)(21,113,33,101)(22,114,34,102)(23,115,35,103)(24,116,36,104)(25,117,37,105)(26,118,38,106)(27,119,39,107)(28,120,40,108)(29,111,31,109)(30,112,32,110)(51,158,78,129)(52,159,79,130)(53,160,80,121)(54,151,71,122)(55,152,72,123)(56,153,73,124)(57,154,74,125)(58,155,75,126)(59,156,76,127)(60,157,77,128), (1,13,28,56)(2,81,29,74)(3,15,30,58)(4,83,21,76)(5,17,22,60)(6,85,23,78)(7,19,24,52)(8,87,25,80)(9,11,26,54)(10,89,27,72)(12,39,55,62)(14,31,57,64)(16,33,59,66)(18,35,51,68)(20,37,53,70)(32,75,65,82)(34,77,67,84)(36,79,69,86)(38,71,61,88)(40,73,63,90)(41,103,158,136)(42,116,159,149)(43,105,160,138)(44,118,151,141)(45,107,152,140)(46,120,153,143)(47,109,154,132)(48,112,155,145)(49,101,156,134)(50,114,157,147)(91,106,122,139)(92,119,123,142)(93,108,124,131)(94,111,125,144)(95,110,126,133)(96,113,127,146)(97,102,128,135)(98,115,129,148)(99,104,130,137)(100,117,121,150), (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), (1,102,63,114)(2,113,64,101)(3,110,65,112)(4,111,66,109)(5,108,67,120)(6,119,68,107)(7,106,69,118)(8,117,70,105)(9,104,61,116)(10,115,62,103)(11,159,88,130)(12,129,89,158)(13,157,90,128)(14,127,81,156)(15,155,82,126)(16,125,83,154)(17,153,84,124)(18,123,85,152)(19,151,86,122)(20,121,87,160)(21,144,33,132)(22,131,34,143)(23,142,35,140)(24,139,36,141)(25,150,37,138)(26,137,38,149)(27,148,39,136)(28,135,40,147)(29,146,31,134)(30,133,32,145)(41,55,98,72)(42,71,99,54)(43,53,100,80)(44,79,91,52)(45,51,92,78)(46,77,93,60)(47,59,94,76)(48,75,95,58)(49,57,96,74)(50,73,97,56)>;
G:=Group( (1,143,63,131)(2,144,64,132)(3,145,65,133)(4,146,66,134)(5,147,67,135)(6,148,68,136)(7,149,69,137)(8,150,70,138)(9,141,61,139)(10,142,62,140)(11,44,88,91)(12,45,89,92)(13,46,90,93)(14,47,81,94)(15,48,82,95)(16,49,83,96)(17,50,84,97)(18,41,85,98)(19,42,86,99)(20,43,87,100)(21,113,33,101)(22,114,34,102)(23,115,35,103)(24,116,36,104)(25,117,37,105)(26,118,38,106)(27,119,39,107)(28,120,40,108)(29,111,31,109)(30,112,32,110)(51,158,78,129)(52,159,79,130)(53,160,80,121)(54,151,71,122)(55,152,72,123)(56,153,73,124)(57,154,74,125)(58,155,75,126)(59,156,76,127)(60,157,77,128), (1,13,28,56)(2,81,29,74)(3,15,30,58)(4,83,21,76)(5,17,22,60)(6,85,23,78)(7,19,24,52)(8,87,25,80)(9,11,26,54)(10,89,27,72)(12,39,55,62)(14,31,57,64)(16,33,59,66)(18,35,51,68)(20,37,53,70)(32,75,65,82)(34,77,67,84)(36,79,69,86)(38,71,61,88)(40,73,63,90)(41,103,158,136)(42,116,159,149)(43,105,160,138)(44,118,151,141)(45,107,152,140)(46,120,153,143)(47,109,154,132)(48,112,155,145)(49,101,156,134)(50,114,157,147)(91,106,122,139)(92,119,123,142)(93,108,124,131)(94,111,125,144)(95,110,126,133)(96,113,127,146)(97,102,128,135)(98,115,129,148)(99,104,130,137)(100,117,121,150), (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), (1,102,63,114)(2,113,64,101)(3,110,65,112)(4,111,66,109)(5,108,67,120)(6,119,68,107)(7,106,69,118)(8,117,70,105)(9,104,61,116)(10,115,62,103)(11,159,88,130)(12,129,89,158)(13,157,90,128)(14,127,81,156)(15,155,82,126)(16,125,83,154)(17,153,84,124)(18,123,85,152)(19,151,86,122)(20,121,87,160)(21,144,33,132)(22,131,34,143)(23,142,35,140)(24,139,36,141)(25,150,37,138)(26,137,38,149)(27,148,39,136)(28,135,40,147)(29,146,31,134)(30,133,32,145)(41,55,98,72)(42,71,99,54)(43,53,100,80)(44,79,91,52)(45,51,92,78)(46,77,93,60)(47,59,94,76)(48,75,95,58)(49,57,96,74)(50,73,97,56) );
G=PermutationGroup([(1,143,63,131),(2,144,64,132),(3,145,65,133),(4,146,66,134),(5,147,67,135),(6,148,68,136),(7,149,69,137),(8,150,70,138),(9,141,61,139),(10,142,62,140),(11,44,88,91),(12,45,89,92),(13,46,90,93),(14,47,81,94),(15,48,82,95),(16,49,83,96),(17,50,84,97),(18,41,85,98),(19,42,86,99),(20,43,87,100),(21,113,33,101),(22,114,34,102),(23,115,35,103),(24,116,36,104),(25,117,37,105),(26,118,38,106),(27,119,39,107),(28,120,40,108),(29,111,31,109),(30,112,32,110),(51,158,78,129),(52,159,79,130),(53,160,80,121),(54,151,71,122),(55,152,72,123),(56,153,73,124),(57,154,74,125),(58,155,75,126),(59,156,76,127),(60,157,77,128)], [(1,13,28,56),(2,81,29,74),(3,15,30,58),(4,83,21,76),(5,17,22,60),(6,85,23,78),(7,19,24,52),(8,87,25,80),(9,11,26,54),(10,89,27,72),(12,39,55,62),(14,31,57,64),(16,33,59,66),(18,35,51,68),(20,37,53,70),(32,75,65,82),(34,77,67,84),(36,79,69,86),(38,71,61,88),(40,73,63,90),(41,103,158,136),(42,116,159,149),(43,105,160,138),(44,118,151,141),(45,107,152,140),(46,120,153,143),(47,109,154,132),(48,112,155,145),(49,101,156,134),(50,114,157,147),(91,106,122,139),(92,119,123,142),(93,108,124,131),(94,111,125,144),(95,110,126,133),(96,113,127,146),(97,102,128,135),(98,115,129,148),(99,104,130,137),(100,117,121,150)], [(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)], [(1,102,63,114),(2,113,64,101),(3,110,65,112),(4,111,66,109),(5,108,67,120),(6,119,68,107),(7,106,69,118),(8,117,70,105),(9,104,61,116),(10,115,62,103),(11,159,88,130),(12,129,89,158),(13,157,90,128),(14,127,81,156),(15,155,82,126),(16,125,83,154),(17,153,84,124),(18,123,85,152),(19,151,86,122),(20,121,87,160),(21,144,33,132),(22,131,34,143),(23,142,35,140),(24,139,36,141),(25,150,37,138),(26,137,38,149),(27,148,39,136),(28,135,40,147),(29,146,31,134),(30,133,32,145),(41,55,98,72),(42,71,99,54),(43,53,100,80),(44,79,91,52),(45,51,92,78),(46,77,93,60),(47,59,94,76),(48,75,95,58),(49,57,96,74),(50,73,97,56)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀