metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.160D10, C10.992- (1+4), C20⋊Q8⋊40C2, C4⋊C4.117D10, C42⋊D5⋊5C2, C42⋊2C2⋊1D5, D10⋊Q8⋊39C2, (C4×Dic10)⋊14C2, (C4×C20).32C22, C22⋊C4.75D10, Dic5⋊3Q8⋊40C2, (C2×C20).192C23, (C2×C10).246C24, Dic5⋊4D4.4C2, C23.52(C22×D5), Dic5.47(C4○D4), C23.D10⋊43C2, C4⋊Dic5.317C22, (C22×C10).60C23, Dic5.5D4.4C2, C22.267(C23×D5), Dic5.14D4⋊44C2, C23.D5.62C22, C23.11D10⋊20C2, C5⋊7(C22.50C24), (C2×Dic5).273C23, (C4×Dic5).157C22, (C22×D5).110C23, C2.63(D4.10D10), D10⋊C4.139C22, (C2×Dic10).262C22, C10.D4.145C22, (C22×Dic5).149C22, C2.93(D5×C4○D4), C4⋊C4⋊D5⋊39C2, (C5×C42⋊2C2)⋊1C2, C10.204(C2×C4○D4), (C2×C4×D5).269C22, (C2×C4).83(C22×D5), (C5×C4⋊C4).201C22, (C2×C5⋊D4).67C22, (C5×C22⋊C4).71C22, SmallGroup(320,1374)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 678 in 212 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×2], C4 [×15], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, D5, C10 [×3], C10, C42, C42 [×6], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic5 [×4], Dic5 [×5], C20 [×6], D10 [×3], C2×C10, C2×C10 [×3], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8, Dic10 [×6], C4×D5 [×2], C2×Dic5 [×7], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×6], C22×D5, C22×C10, C22.50C24, C4×Dic5 [×6], C10.D4 [×7], C4⋊Dic5 [×2], D10⋊C4 [×5], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×Dic10, C42⋊D5, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊4D4, Dic5.5D4 [×2], Dic5⋊3Q8 [×2], C20⋊Q8, D10⋊Q8, C4⋊C4⋊D5 [×2], C5×C42⋊2C2, C42.160D10
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], C22.50C24, C23×D5, D5×C4○D4 [×2], D4.10D10, C42.160D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 115 95 32)(2 71 96 144)(3 117 97 34)(4 73 98 146)(5 119 99 36)(6 75 100 148)(7 111 91 38)(8 77 92 150)(9 113 93 40)(10 79 94 142)(11 69 41 132)(12 105 42 22)(13 61 43 134)(14 107 44 24)(15 63 45 136)(16 109 46 26)(17 65 47 138)(18 101 48 28)(19 67 49 140)(20 103 50 30)(21 85 104 152)(23 87 106 154)(25 89 108 156)(27 81 110 158)(29 83 102 160)(31 122 114 59)(33 124 116 51)(35 126 118 53)(37 128 120 55)(39 130 112 57)(52 145 125 72)(54 147 127 74)(56 149 129 76)(58 141 121 78)(60 143 123 80)(62 155 135 88)(64 157 137 90)(66 159 139 82)(68 151 131 84)(70 153 133 86)
(1 133 123 22)(2 106 124 61)(3 135 125 24)(4 108 126 63)(5 137 127 26)(6 110 128 65)(7 139 129 28)(8 102 130 67)(9 131 121 30)(10 104 122 69)(11 142 85 31)(12 115 86 80)(13 144 87 33)(14 117 88 72)(15 146 89 35)(16 119 90 74)(17 148 81 37)(18 111 82 76)(19 150 83 39)(20 113 84 78)(21 59 132 94)(23 51 134 96)(25 53 136 98)(27 55 138 100)(29 57 140 92)(32 153 143 42)(34 155 145 44)(36 157 147 46)(38 159 149 48)(40 151 141 50)(41 79 152 114)(43 71 154 116)(45 73 156 118)(47 75 158 120)(49 77 160 112)(52 107 97 62)(54 109 99 64)(56 101 91 66)(58 103 93 68)(60 105 95 70)
(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 21 123 132)(2 30 124 131)(3 29 125 140)(4 28 126 139)(5 27 127 138)(6 26 128 137)(7 25 129 136)(8 24 130 135)(9 23 121 134)(10 22 122 133)(11 115 85 80)(12 114 86 79)(13 113 87 78)(14 112 88 77)(15 111 89 76)(16 120 90 75)(17 119 81 74)(18 118 82 73)(19 117 83 72)(20 116 84 71)(31 153 142 42)(32 152 143 41)(33 151 144 50)(34 160 145 49)(35 159 146 48)(36 158 147 47)(37 157 148 46)(38 156 149 45)(39 155 150 44)(40 154 141 43)(51 68 96 103)(52 67 97 102)(53 66 98 101)(54 65 99 110)(55 64 100 109)(56 63 91 108)(57 62 92 107)(58 61 93 106)(59 70 94 105)(60 69 95 104)
G:=sub<Sym(160)| (1,115,95,32)(2,71,96,144)(3,117,97,34)(4,73,98,146)(5,119,99,36)(6,75,100,148)(7,111,91,38)(8,77,92,150)(9,113,93,40)(10,79,94,142)(11,69,41,132)(12,105,42,22)(13,61,43,134)(14,107,44,24)(15,63,45,136)(16,109,46,26)(17,65,47,138)(18,101,48,28)(19,67,49,140)(20,103,50,30)(21,85,104,152)(23,87,106,154)(25,89,108,156)(27,81,110,158)(29,83,102,160)(31,122,114,59)(33,124,116,51)(35,126,118,53)(37,128,120,55)(39,130,112,57)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,22)(2,106,124,61)(3,135,125,24)(4,108,126,63)(5,137,127,26)(6,110,128,65)(7,139,129,28)(8,102,130,67)(9,131,121,30)(10,104,122,69)(11,142,85,31)(12,115,86,80)(13,144,87,33)(14,117,88,72)(15,146,89,35)(16,119,90,74)(17,148,81,37)(18,111,82,76)(19,150,83,39)(20,113,84,78)(21,59,132,94)(23,51,134,96)(25,53,136,98)(27,55,138,100)(29,57,140,92)(32,153,143,42)(34,155,145,44)(36,157,147,46)(38,159,149,48)(40,151,141,50)(41,79,152,114)(43,71,154,116)(45,73,156,118)(47,75,158,120)(49,77,160,112)(52,107,97,62)(54,109,99,64)(56,101,91,66)(58,103,93,68)(60,105,95,70), (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), (1,21,123,132)(2,30,124,131)(3,29,125,140)(4,28,126,139)(5,27,127,138)(6,26,128,137)(7,25,129,136)(8,24,130,135)(9,23,121,134)(10,22,122,133)(11,115,85,80)(12,114,86,79)(13,113,87,78)(14,112,88,77)(15,111,89,76)(16,120,90,75)(17,119,81,74)(18,118,82,73)(19,117,83,72)(20,116,84,71)(31,153,142,42)(32,152,143,41)(33,151,144,50)(34,160,145,49)(35,159,146,48)(36,158,147,47)(37,157,148,46)(38,156,149,45)(39,155,150,44)(40,154,141,43)(51,68,96,103)(52,67,97,102)(53,66,98,101)(54,65,99,110)(55,64,100,109)(56,63,91,108)(57,62,92,107)(58,61,93,106)(59,70,94,105)(60,69,95,104)>;
G:=Group( (1,115,95,32)(2,71,96,144)(3,117,97,34)(4,73,98,146)(5,119,99,36)(6,75,100,148)(7,111,91,38)(8,77,92,150)(9,113,93,40)(10,79,94,142)(11,69,41,132)(12,105,42,22)(13,61,43,134)(14,107,44,24)(15,63,45,136)(16,109,46,26)(17,65,47,138)(18,101,48,28)(19,67,49,140)(20,103,50,30)(21,85,104,152)(23,87,106,154)(25,89,108,156)(27,81,110,158)(29,83,102,160)(31,122,114,59)(33,124,116,51)(35,126,118,53)(37,128,120,55)(39,130,112,57)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,22)(2,106,124,61)(3,135,125,24)(4,108,126,63)(5,137,127,26)(6,110,128,65)(7,139,129,28)(8,102,130,67)(9,131,121,30)(10,104,122,69)(11,142,85,31)(12,115,86,80)(13,144,87,33)(14,117,88,72)(15,146,89,35)(16,119,90,74)(17,148,81,37)(18,111,82,76)(19,150,83,39)(20,113,84,78)(21,59,132,94)(23,51,134,96)(25,53,136,98)(27,55,138,100)(29,57,140,92)(32,153,143,42)(34,155,145,44)(36,157,147,46)(38,159,149,48)(40,151,141,50)(41,79,152,114)(43,71,154,116)(45,73,156,118)(47,75,158,120)(49,77,160,112)(52,107,97,62)(54,109,99,64)(56,101,91,66)(58,103,93,68)(60,105,95,70), (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), (1,21,123,132)(2,30,124,131)(3,29,125,140)(4,28,126,139)(5,27,127,138)(6,26,128,137)(7,25,129,136)(8,24,130,135)(9,23,121,134)(10,22,122,133)(11,115,85,80)(12,114,86,79)(13,113,87,78)(14,112,88,77)(15,111,89,76)(16,120,90,75)(17,119,81,74)(18,118,82,73)(19,117,83,72)(20,116,84,71)(31,153,142,42)(32,152,143,41)(33,151,144,50)(34,160,145,49)(35,159,146,48)(36,158,147,47)(37,157,148,46)(38,156,149,45)(39,155,150,44)(40,154,141,43)(51,68,96,103)(52,67,97,102)(53,66,98,101)(54,65,99,110)(55,64,100,109)(56,63,91,108)(57,62,92,107)(58,61,93,106)(59,70,94,105)(60,69,95,104) );
G=PermutationGroup([(1,115,95,32),(2,71,96,144),(3,117,97,34),(4,73,98,146),(5,119,99,36),(6,75,100,148),(7,111,91,38),(8,77,92,150),(9,113,93,40),(10,79,94,142),(11,69,41,132),(12,105,42,22),(13,61,43,134),(14,107,44,24),(15,63,45,136),(16,109,46,26),(17,65,47,138),(18,101,48,28),(19,67,49,140),(20,103,50,30),(21,85,104,152),(23,87,106,154),(25,89,108,156),(27,81,110,158),(29,83,102,160),(31,122,114,59),(33,124,116,51),(35,126,118,53),(37,128,120,55),(39,130,112,57),(52,145,125,72),(54,147,127,74),(56,149,129,76),(58,141,121,78),(60,143,123,80),(62,155,135,88),(64,157,137,90),(66,159,139,82),(68,151,131,84),(70,153,133,86)], [(1,133,123,22),(2,106,124,61),(3,135,125,24),(4,108,126,63),(5,137,127,26),(6,110,128,65),(7,139,129,28),(8,102,130,67),(9,131,121,30),(10,104,122,69),(11,142,85,31),(12,115,86,80),(13,144,87,33),(14,117,88,72),(15,146,89,35),(16,119,90,74),(17,148,81,37),(18,111,82,76),(19,150,83,39),(20,113,84,78),(21,59,132,94),(23,51,134,96),(25,53,136,98),(27,55,138,100),(29,57,140,92),(32,153,143,42),(34,155,145,44),(36,157,147,46),(38,159,149,48),(40,151,141,50),(41,79,152,114),(43,71,154,116),(45,73,156,118),(47,75,158,120),(49,77,160,112),(52,107,97,62),(54,109,99,64),(56,101,91,66),(58,103,93,68),(60,105,95,70)], [(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)], [(1,21,123,132),(2,30,124,131),(3,29,125,140),(4,28,126,139),(5,27,127,138),(6,26,128,137),(7,25,129,136),(8,24,130,135),(9,23,121,134),(10,22,122,133),(11,115,85,80),(12,114,86,79),(13,113,87,78),(14,112,88,77),(15,111,89,76),(16,120,90,75),(17,119,81,74),(18,118,82,73),(19,117,83,72),(20,116,84,71),(31,153,142,42),(32,152,143,41),(33,151,144,50),(34,160,145,49),(35,159,146,48),(36,158,147,47),(37,157,148,46),(38,156,149,45),(39,155,150,44),(40,154,141,43),(51,68,96,103),(52,67,97,102),(53,66,98,101),(54,65,99,110),(55,64,100,109),(56,63,91,108),(57,62,92,107),(58,61,93,106),(59,70,94,105),(60,69,95,104)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 33 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 | 0 | 0 |
| 0 | 0 | 30 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 7 | 35 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 | 0 | 0 |
| 0 | 0 | 30 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,33,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,8,30,0,0,0,0,2,33,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[7,8,0,0,0,0,35,34,0,0,0,0,0,0,8,30,0,0,0,0,2,33,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- (1+4) | D5×C4○D4 | D4.10D10 |
| kernel | C42.160D10 | C4×Dic10 | C42⋊D5 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | Dic5.5D4 | Dic5⋊3Q8 | C20⋊Q8 | D10⋊Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 8 | 2 | 6 | 6 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{160}D_{10} % in TeX
G:=Group("C4^2.160D10"); // GroupNames label
G:=SmallGroup(320,1374);
// by ID
G=gap.SmallGroup(320,1374);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,387,100,794,136,12550]);
// 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=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀