metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.166D10, C10.752+ (1+4), C4⋊1D4.7D5, (D4×Dic5)⋊33C2, (C4×Dic10)⋊50C2, (C2×D4).114D10, C20.133(C4○D4), C4.17(D4⋊2D5), C20.17D4⋊25C2, (C4×C20).202C22, (C2×C20).634C23, (C2×C10).257C24, C2.79(D4⋊6D10), C23.63(C22×D5), (D4×C10).160C22, C4⋊Dic5.380C22, (C22×C10).71C23, C22.278(C23×D5), C23.D5.71C22, C23.18D10⋊26C2, C5⋊5(C22.53C24), (C4×Dic5).162C22, (C2×Dic5).133C23, (C2×Dic10).308C22, C10.D4.163C22, (C22×Dic5).156C22, C10.95(C2×C4○D4), (C5×C4⋊1D4).6C2, C2.59(C2×D4⋊2D5), (C2×C4).595(C22×D5), SmallGroup(320,1385)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 726 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×10], Q8 [×4], C23 [×4], C10, C10 [×2], C10 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], Dic5 [×8], C20 [×4], C20, C2×C10, C2×C10 [×12], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C4⋊1D4, Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C5×D4 [×10], C22×C10 [×4], C22.53C24, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×2], C23.D5 [×12], C4×C20, C2×Dic10 [×2], C22×Dic5 [×4], D4×C10 [×6], C4×Dic10 [×2], D4×Dic5 [×4], C23.18D10 [×4], C20.17D4 [×4], C5×C4⋊1D4, C42.166D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.53C24, D4⋊2D5 [×4], C23×D5, C2×D4⋊2D5 [×2], D4⋊6D10, C42.166D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 56 51 6)(2 7 52 57)(3 58 53 8)(4 9 54 59)(5 60 55 10)(11 89 21 71)(12 72 22 90)(13 81 23 73)(14 74 24 82)(15 83 25 75)(16 76 26 84)(17 85 27 77)(18 78 28 86)(19 87 29 79)(20 80 30 88)(31 36 64 69)(32 70 65 37)(33 38 66 61)(34 62 67 39)(35 40 68 63)(41 130 158 103)(42 104 159 121)(43 122 160 105)(44 106 151 123)(45 124 152 107)(46 108 153 125)(47 126 154 109)(48 110 155 127)(49 128 156 101)(50 102 157 129)(91 138 133 96)(92 97 134 139)(93 140 135 98)(94 99 136 131)(95 132 137 100)(111 148 143 116)(112 117 144 149)(113 150 145 118)(114 119 146 141)(115 142 147 120)
(1 73 63 28)(2 29 64 74)(3 75 65 30)(4 21 66 76)(5 77 67 22)(6 23 68 78)(7 79 69 24)(8 25 70 80)(9 71 61 26)(10 27 62 72)(11 33 84 54)(12 55 85 34)(13 35 86 56)(14 57 87 36)(15 37 88 58)(16 59 89 38)(17 39 90 60)(18 51 81 40)(19 31 82 52)(20 53 83 32)(41 148 125 98)(42 99 126 149)(43 150 127 100)(44 91 128 141)(45 142 129 92)(46 93 130 143)(47 144 121 94)(48 95 122 145)(49 146 123 96)(50 97 124 147)(101 119 151 133)(102 134 152 120)(103 111 153 135)(104 136 154 112)(105 113 155 137)(106 138 156 114)(107 115 157 139)(108 140 158 116)(109 117 159 131)(110 132 160 118)
(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)
(1 146 63 96)(2 145 64 95)(3 144 65 94)(4 143 66 93)(5 142 67 92)(6 141 68 91)(7 150 69 100)(8 149 70 99)(9 148 61 98)(10 147 62 97)(11 46 84 130)(12 45 85 129)(13 44 86 128)(14 43 87 127)(15 42 88 126)(16 41 89 125)(17 50 90 124)(18 49 81 123)(19 48 82 122)(20 47 83 121)(21 153 76 103)(22 152 77 102)(23 151 78 101)(24 160 79 110)(25 159 80 109)(26 158 71 108)(27 157 72 107)(28 156 73 106)(29 155 74 105)(30 154 75 104)(31 137 52 113)(32 136 53 112)(33 135 54 111)(34 134 55 120)(35 133 56 119)(36 132 57 118)(37 131 58 117)(38 140 59 116)(39 139 60 115)(40 138 51 114)
G:=sub<Sym(160)| (1,56,51,6)(2,7,52,57)(3,58,53,8)(4,9,54,59)(5,60,55,10)(11,89,21,71)(12,72,22,90)(13,81,23,73)(14,74,24,82)(15,83,25,75)(16,76,26,84)(17,85,27,77)(18,78,28,86)(19,87,29,79)(20,80,30,88)(31,36,64,69)(32,70,65,37)(33,38,66,61)(34,62,67,39)(35,40,68,63)(41,130,158,103)(42,104,159,121)(43,122,160,105)(44,106,151,123)(45,124,152,107)(46,108,153,125)(47,126,154,109)(48,110,155,127)(49,128,156,101)(50,102,157,129)(91,138,133,96)(92,97,134,139)(93,140,135,98)(94,99,136,131)(95,132,137,100)(111,148,143,116)(112,117,144,149)(113,150,145,118)(114,119,146,141)(115,142,147,120), (1,73,63,28)(2,29,64,74)(3,75,65,30)(4,21,66,76)(5,77,67,22)(6,23,68,78)(7,79,69,24)(8,25,70,80)(9,71,61,26)(10,27,62,72)(11,33,84,54)(12,55,85,34)(13,35,86,56)(14,57,87,36)(15,37,88,58)(16,59,89,38)(17,39,90,60)(18,51,81,40)(19,31,82,52)(20,53,83,32)(41,148,125,98)(42,99,126,149)(43,150,127,100)(44,91,128,141)(45,142,129,92)(46,93,130,143)(47,144,121,94)(48,95,122,145)(49,146,123,96)(50,97,124,147)(101,119,151,133)(102,134,152,120)(103,111,153,135)(104,136,154,112)(105,113,155,137)(106,138,156,114)(107,115,157,139)(108,140,158,116)(109,117,159,131)(110,132,160,118), (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), (1,146,63,96)(2,145,64,95)(3,144,65,94)(4,143,66,93)(5,142,67,92)(6,141,68,91)(7,150,69,100)(8,149,70,99)(9,148,61,98)(10,147,62,97)(11,46,84,130)(12,45,85,129)(13,44,86,128)(14,43,87,127)(15,42,88,126)(16,41,89,125)(17,50,90,124)(18,49,81,123)(19,48,82,122)(20,47,83,121)(21,153,76,103)(22,152,77,102)(23,151,78,101)(24,160,79,110)(25,159,80,109)(26,158,71,108)(27,157,72,107)(28,156,73,106)(29,155,74,105)(30,154,75,104)(31,137,52,113)(32,136,53,112)(33,135,54,111)(34,134,55,120)(35,133,56,119)(36,132,57,118)(37,131,58,117)(38,140,59,116)(39,139,60,115)(40,138,51,114)>;
G:=Group( (1,56,51,6)(2,7,52,57)(3,58,53,8)(4,9,54,59)(5,60,55,10)(11,89,21,71)(12,72,22,90)(13,81,23,73)(14,74,24,82)(15,83,25,75)(16,76,26,84)(17,85,27,77)(18,78,28,86)(19,87,29,79)(20,80,30,88)(31,36,64,69)(32,70,65,37)(33,38,66,61)(34,62,67,39)(35,40,68,63)(41,130,158,103)(42,104,159,121)(43,122,160,105)(44,106,151,123)(45,124,152,107)(46,108,153,125)(47,126,154,109)(48,110,155,127)(49,128,156,101)(50,102,157,129)(91,138,133,96)(92,97,134,139)(93,140,135,98)(94,99,136,131)(95,132,137,100)(111,148,143,116)(112,117,144,149)(113,150,145,118)(114,119,146,141)(115,142,147,120), (1,73,63,28)(2,29,64,74)(3,75,65,30)(4,21,66,76)(5,77,67,22)(6,23,68,78)(7,79,69,24)(8,25,70,80)(9,71,61,26)(10,27,62,72)(11,33,84,54)(12,55,85,34)(13,35,86,56)(14,57,87,36)(15,37,88,58)(16,59,89,38)(17,39,90,60)(18,51,81,40)(19,31,82,52)(20,53,83,32)(41,148,125,98)(42,99,126,149)(43,150,127,100)(44,91,128,141)(45,142,129,92)(46,93,130,143)(47,144,121,94)(48,95,122,145)(49,146,123,96)(50,97,124,147)(101,119,151,133)(102,134,152,120)(103,111,153,135)(104,136,154,112)(105,113,155,137)(106,138,156,114)(107,115,157,139)(108,140,158,116)(109,117,159,131)(110,132,160,118), (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), (1,146,63,96)(2,145,64,95)(3,144,65,94)(4,143,66,93)(5,142,67,92)(6,141,68,91)(7,150,69,100)(8,149,70,99)(9,148,61,98)(10,147,62,97)(11,46,84,130)(12,45,85,129)(13,44,86,128)(14,43,87,127)(15,42,88,126)(16,41,89,125)(17,50,90,124)(18,49,81,123)(19,48,82,122)(20,47,83,121)(21,153,76,103)(22,152,77,102)(23,151,78,101)(24,160,79,110)(25,159,80,109)(26,158,71,108)(27,157,72,107)(28,156,73,106)(29,155,74,105)(30,154,75,104)(31,137,52,113)(32,136,53,112)(33,135,54,111)(34,134,55,120)(35,133,56,119)(36,132,57,118)(37,131,58,117)(38,140,59,116)(39,139,60,115)(40,138,51,114) );
G=PermutationGroup([(1,56,51,6),(2,7,52,57),(3,58,53,8),(4,9,54,59),(5,60,55,10),(11,89,21,71),(12,72,22,90),(13,81,23,73),(14,74,24,82),(15,83,25,75),(16,76,26,84),(17,85,27,77),(18,78,28,86),(19,87,29,79),(20,80,30,88),(31,36,64,69),(32,70,65,37),(33,38,66,61),(34,62,67,39),(35,40,68,63),(41,130,158,103),(42,104,159,121),(43,122,160,105),(44,106,151,123),(45,124,152,107),(46,108,153,125),(47,126,154,109),(48,110,155,127),(49,128,156,101),(50,102,157,129),(91,138,133,96),(92,97,134,139),(93,140,135,98),(94,99,136,131),(95,132,137,100),(111,148,143,116),(112,117,144,149),(113,150,145,118),(114,119,146,141),(115,142,147,120)], [(1,73,63,28),(2,29,64,74),(3,75,65,30),(4,21,66,76),(5,77,67,22),(6,23,68,78),(7,79,69,24),(8,25,70,80),(9,71,61,26),(10,27,62,72),(11,33,84,54),(12,55,85,34),(13,35,86,56),(14,57,87,36),(15,37,88,58),(16,59,89,38),(17,39,90,60),(18,51,81,40),(19,31,82,52),(20,53,83,32),(41,148,125,98),(42,99,126,149),(43,150,127,100),(44,91,128,141),(45,142,129,92),(46,93,130,143),(47,144,121,94),(48,95,122,145),(49,146,123,96),(50,97,124,147),(101,119,151,133),(102,134,152,120),(103,111,153,135),(104,136,154,112),(105,113,155,137),(106,138,156,114),(107,115,157,139),(108,140,158,116),(109,117,159,131),(110,132,160,118)], [(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)], [(1,146,63,96),(2,145,64,95),(3,144,65,94),(4,143,66,93),(5,142,67,92),(6,141,68,91),(7,150,69,100),(8,149,70,99),(9,148,61,98),(10,147,62,97),(11,46,84,130),(12,45,85,129),(13,44,86,128),(14,43,87,127),(15,42,88,126),(16,41,89,125),(17,50,90,124),(18,49,81,123),(19,48,82,122),(20,47,83,121),(21,153,76,103),(22,152,77,102),(23,151,78,101),(24,160,79,110),(25,159,80,109),(26,158,71,108),(27,157,72,107),(28,156,73,106),(29,155,74,105),(30,154,75,104),(31,137,52,113),(32,136,53,112),(33,135,54,111),(34,134,55,120),(35,133,56,119),(36,132,57,118),(37,131,58,117),(38,140,59,116),(39,139,60,115),(40,138,51,114)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 25 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,2,13,0,0,0,0,25,39,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D4⋊6D10 |
| kernel | C42.166D10 | C4×Dic10 | D4×Dic5 | C23.18D10 | C20.17D4 | C5×C4⋊1D4 | C4⋊1D4 | C20 | C42 | C2×D4 | C10 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 8 | 2 | 12 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{166}D_{10} % in TeX
G:=Group("C4^2.166D10"); // GroupNames label
G:=SmallGroup(320,1385);
// by ID
G=gap.SmallGroup(320,1385);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^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=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀