metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.118D10, C10.232+ (1+4), C10.632- (1+4), (C4×D4)⋊26D5, (D4×C20)⋊28C2, C4⋊C4.288D10, D10⋊Q8⋊9C2, (C2×D4).225D10, C42⋊2D5⋊11C2, C20.6Q8⋊26C2, (C22×C4).49D10, C20.48D4⋊13C2, (C4×C20).220C22, (C2×C20).166C23, (C2×C10).108C24, C22⋊C4.120D10, Dic5.Q8⋊8C2, Dic5⋊D4.4C2, C22.7(C4○D20), C22.D20⋊7C2, C4⋊Dic5.41C22, D10.12D4⋊10C2, C2.25(D4⋊6D10), (D4×C10).309C22, C23.D10⋊10C2, C23.23D10⋊5C2, (C2×Dic5).48C23, C10.D4.8C22, (C22×D5).42C23, C23.105(C22×D5), C22.133(C23×D5), Dic5.14D4⋊10C2, C23.D5.17C22, D10⋊C4.67C22, C23.18D10⋊19C2, (C22×C20).366C22, (C22×C10).178C23, C5⋊2(C22.33C24), (C2×Dic10).31C22, (C4×Dic5).227C22, C2.20(D4.10D10), (C22×Dic5).100C22, (C4×C5⋊D4)⋊47C2, C2.57(C2×C4○D20), C10.50(C2×C4○D4), (C2×C4×D5).255C22, (C2×C10).18(C4○D4), (C2×C10.D4)⋊39C2, (C5×C4⋊C4).336C22, (C2×C4).164(C22×D5), (C2×C5⋊D4).125C22, (C5×C22⋊C4).107C22, SmallGroup(320,1236)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 718 in 218 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23, D5, C10 [×3], C10 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×13], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×7], C20 [×5], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], Dic10, C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×C10 [×2], C22.33C24, C4×Dic5, C10.D4 [×10], C4⋊Dic5 [×3], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C20.6Q8, C42⋊2D5, Dic5.14D4, C23.D10, D10.12D4, C22.D20, Dic5.Q8, D10⋊Q8, C2×C10.D4, C20.48D4, C4×C5⋊D4, C23.23D10, C23.18D10, Dic5⋊D4, D4×C20, C42.118D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.33C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊6D10, D4.10D10, C42.118D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >
►On 160 points
Generators in S160
(1 138 123 21)(2 22 124 139)(3 140 125 23)(4 24 126 131)(5 132 127 25)(6 26 128 133)(7 134 129 27)(8 28 130 135)(9 136 121 29)(10 30 122 137)(11 80 81 119)(12 120 82 71)(13 72 83 111)(14 112 84 73)(15 74 85 113)(16 114 86 75)(17 76 87 115)(18 116 88 77)(19 78 89 117)(20 118 90 79)(31 153 148 46)(32 47 149 154)(33 155 150 48)(34 49 141 156)(35 157 142 50)(36 41 143 158)(37 159 144 42)(38 43 145 160)(39 151 146 44)(40 45 147 152)(51 66 100 105)(52 106 91 67)(53 68 92 107)(54 108 93 69)(55 70 94 109)(56 110 95 61)(57 62 96 101)(58 102 97 63)(59 64 98 103)(60 104 99 65)
(1 86 60 46)(2 17 51 154)(3 88 52 48)(4 19 53 156)(5 90 54 50)(6 11 55 158)(7 82 56 42)(8 13 57 160)(9 84 58 44)(10 15 59 152)(12 95 159 129)(14 97 151 121)(16 99 153 123)(18 91 155 125)(20 93 157 127)(21 114 65 148)(22 76 66 32)(23 116 67 150)(24 78 68 34)(25 118 69 142)(26 80 70 36)(27 120 61 144)(28 72 62 38)(29 112 63 146)(30 74 64 40)(31 138 75 104)(33 140 77 106)(35 132 79 108)(37 134 71 110)(39 136 73 102)(41 128 81 94)(43 130 83 96)(45 122 85 98)(47 124 87 100)(49 126 89 92)(101 145 135 111)(103 147 137 113)(105 149 139 115)(107 141 131 117)(109 143 133 119)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 157 81 50)(12 49 82 156)(13 155 83 48)(14 47 84 154)(15 153 85 46)(16 45 86 152)(17 151 87 44)(18 43 88 160)(19 159 89 42)(20 41 90 158)(21 64 138 103)(22 102 139 63)(23 62 140 101)(24 110 131 61)(25 70 132 109)(26 108 133 69)(27 68 134 107)(28 106 135 67)(29 66 136 105)(30 104 137 65)(31 147 148 40)(32 39 149 146)(33 145 150 38)(34 37 141 144)(35 143 142 36)(51 97 100 58)(52 57 91 96)(53 95 92 56)(54 55 93 94)(59 99 98 60)(71 117 120 78)(72 77 111 116)(73 115 112 76)(74 75 113 114)(79 119 118 80)
G:=sub<Sym(160)| (1,138,123,21)(2,22,124,139)(3,140,125,23)(4,24,126,131)(5,132,127,25)(6,26,128,133)(7,134,129,27)(8,28,130,135)(9,136,121,29)(10,30,122,137)(11,80,81,119)(12,120,82,71)(13,72,83,111)(14,112,84,73)(15,74,85,113)(16,114,86,75)(17,76,87,115)(18,116,88,77)(19,78,89,117)(20,118,90,79)(31,153,148,46)(32,47,149,154)(33,155,150,48)(34,49,141,156)(35,157,142,50)(36,41,143,158)(37,159,144,42)(38,43,145,160)(39,151,146,44)(40,45,147,152)(51,66,100,105)(52,106,91,67)(53,68,92,107)(54,108,93,69)(55,70,94,109)(56,110,95,61)(57,62,96,101)(58,102,97,63)(59,64,98,103)(60,104,99,65), (1,86,60,46)(2,17,51,154)(3,88,52,48)(4,19,53,156)(5,90,54,50)(6,11,55,158)(7,82,56,42)(8,13,57,160)(9,84,58,44)(10,15,59,152)(12,95,159,129)(14,97,151,121)(16,99,153,123)(18,91,155,125)(20,93,157,127)(21,114,65,148)(22,76,66,32)(23,116,67,150)(24,78,68,34)(25,118,69,142)(26,80,70,36)(27,120,61,144)(28,72,62,38)(29,112,63,146)(30,74,64,40)(31,138,75,104)(33,140,77,106)(35,132,79,108)(37,134,71,110)(39,136,73,102)(41,128,81,94)(43,130,83,96)(45,122,85,98)(47,124,87,100)(49,126,89,92)(101,145,135,111)(103,147,137,113)(105,149,139,115)(107,141,131,117)(109,143,133,119), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,157,81,50)(12,49,82,156)(13,155,83,48)(14,47,84,154)(15,153,85,46)(16,45,86,152)(17,151,87,44)(18,43,88,160)(19,159,89,42)(20,41,90,158)(21,64,138,103)(22,102,139,63)(23,62,140,101)(24,110,131,61)(25,70,132,109)(26,108,133,69)(27,68,134,107)(28,106,135,67)(29,66,136,105)(30,104,137,65)(31,147,148,40)(32,39,149,146)(33,145,150,38)(34,37,141,144)(35,143,142,36)(51,97,100,58)(52,57,91,96)(53,95,92,56)(54,55,93,94)(59,99,98,60)(71,117,120,78)(72,77,111,116)(73,115,112,76)(74,75,113,114)(79,119,118,80)>;
G:=Group( (1,138,123,21)(2,22,124,139)(3,140,125,23)(4,24,126,131)(5,132,127,25)(6,26,128,133)(7,134,129,27)(8,28,130,135)(9,136,121,29)(10,30,122,137)(11,80,81,119)(12,120,82,71)(13,72,83,111)(14,112,84,73)(15,74,85,113)(16,114,86,75)(17,76,87,115)(18,116,88,77)(19,78,89,117)(20,118,90,79)(31,153,148,46)(32,47,149,154)(33,155,150,48)(34,49,141,156)(35,157,142,50)(36,41,143,158)(37,159,144,42)(38,43,145,160)(39,151,146,44)(40,45,147,152)(51,66,100,105)(52,106,91,67)(53,68,92,107)(54,108,93,69)(55,70,94,109)(56,110,95,61)(57,62,96,101)(58,102,97,63)(59,64,98,103)(60,104,99,65), (1,86,60,46)(2,17,51,154)(3,88,52,48)(4,19,53,156)(5,90,54,50)(6,11,55,158)(7,82,56,42)(8,13,57,160)(9,84,58,44)(10,15,59,152)(12,95,159,129)(14,97,151,121)(16,99,153,123)(18,91,155,125)(20,93,157,127)(21,114,65,148)(22,76,66,32)(23,116,67,150)(24,78,68,34)(25,118,69,142)(26,80,70,36)(27,120,61,144)(28,72,62,38)(29,112,63,146)(30,74,64,40)(31,138,75,104)(33,140,77,106)(35,132,79,108)(37,134,71,110)(39,136,73,102)(41,128,81,94)(43,130,83,96)(45,122,85,98)(47,124,87,100)(49,126,89,92)(101,145,135,111)(103,147,137,113)(105,149,139,115)(107,141,131,117)(109,143,133,119), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,157,81,50)(12,49,82,156)(13,155,83,48)(14,47,84,154)(15,153,85,46)(16,45,86,152)(17,151,87,44)(18,43,88,160)(19,159,89,42)(20,41,90,158)(21,64,138,103)(22,102,139,63)(23,62,140,101)(24,110,131,61)(25,70,132,109)(26,108,133,69)(27,68,134,107)(28,106,135,67)(29,66,136,105)(30,104,137,65)(31,147,148,40)(32,39,149,146)(33,145,150,38)(34,37,141,144)(35,143,142,36)(51,97,100,58)(52,57,91,96)(53,95,92,56)(54,55,93,94)(59,99,98,60)(71,117,120,78)(72,77,111,116)(73,115,112,76)(74,75,113,114)(79,119,118,80) );
G=PermutationGroup([(1,138,123,21),(2,22,124,139),(3,140,125,23),(4,24,126,131),(5,132,127,25),(6,26,128,133),(7,134,129,27),(8,28,130,135),(9,136,121,29),(10,30,122,137),(11,80,81,119),(12,120,82,71),(13,72,83,111),(14,112,84,73),(15,74,85,113),(16,114,86,75),(17,76,87,115),(18,116,88,77),(19,78,89,117),(20,118,90,79),(31,153,148,46),(32,47,149,154),(33,155,150,48),(34,49,141,156),(35,157,142,50),(36,41,143,158),(37,159,144,42),(38,43,145,160),(39,151,146,44),(40,45,147,152),(51,66,100,105),(52,106,91,67),(53,68,92,107),(54,108,93,69),(55,70,94,109),(56,110,95,61),(57,62,96,101),(58,102,97,63),(59,64,98,103),(60,104,99,65)], [(1,86,60,46),(2,17,51,154),(3,88,52,48),(4,19,53,156),(5,90,54,50),(6,11,55,158),(7,82,56,42),(8,13,57,160),(9,84,58,44),(10,15,59,152),(12,95,159,129),(14,97,151,121),(16,99,153,123),(18,91,155,125),(20,93,157,127),(21,114,65,148),(22,76,66,32),(23,116,67,150),(24,78,68,34),(25,118,69,142),(26,80,70,36),(27,120,61,144),(28,72,62,38),(29,112,63,146),(30,74,64,40),(31,138,75,104),(33,140,77,106),(35,132,79,108),(37,134,71,110),(39,136,73,102),(41,128,81,94),(43,130,83,96),(45,122,85,98),(47,124,87,100),(49,126,89,92),(101,145,135,111),(103,147,137,113),(105,149,139,115),(107,141,131,117),(109,143,133,119)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,157,81,50),(12,49,82,156),(13,155,83,48),(14,47,84,154),(15,153,85,46),(16,45,86,152),(17,151,87,44),(18,43,88,160),(19,159,89,42),(20,41,90,158),(21,64,138,103),(22,102,139,63),(23,62,140,101),(24,110,131,61),(25,70,132,109),(26,108,133,69),(27,68,134,107),(28,106,135,67),(29,66,136,105),(30,104,137,65),(31,147,148,40),(32,39,149,146),(33,145,150,38),(34,37,141,144),(35,143,142,36),(51,97,100,58),(52,57,91,96),(53,95,92,56),(54,55,93,94),(59,99,98,60),(71,117,120,78),(72,77,111,116),(73,115,112,76),(74,75,113,114),(79,119,118,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 16 | 18 | 33 |
| 0 | 0 | 17 | 34 | 0 | 33 |
| 0 | 0 | 0 | 25 | 18 | 25 |
| 0 | 0 | 24 | 30 | 28 | 31 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 37 | 13 |
| 0 | 0 | 28 | 30 | 15 | 28 |
| 0 | 0 | 15 | 15 | 11 | 0 |
| 0 | 0 | 20 | 33 | 4 | 39 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 26 | 28 | 4 |
| 0 | 0 | 15 | 15 | 37 | 0 |
| 0 | 0 | 26 | 0 | 8 | 15 |
| 0 | 0 | 30 | 26 | 16 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 13 | 17 |
| 0 | 0 | 19 | 34 | 4 | 4 |
| 0 | 0 | 18 | 25 | 33 | 7 |
| 0 | 0 | 14 | 30 | 25 | 32 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,17,0,24,0,0,16,34,25,30,0,0,18,0,18,28,0,0,33,33,25,31],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,2,28,15,20,0,0,0,30,15,33,0,0,37,15,11,4,0,0,13,28,0,39],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,15,26,30,0,0,26,15,0,26,0,0,28,37,8,16,0,0,4,0,15,7],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,24,19,18,14,0,0,16,34,25,30,0,0,13,4,33,25,0,0,17,4,7,32] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D4.10D10 |
| kernel | C42.118D10 | C20.6Q8 | C42⋊2D5 | Dic5.14D4 | C23.D10 | D10.12D4 | C22.D20 | Dic5.Q8 | D10⋊Q8 | C2×C10.D4 | C20.48D4 | C4×C5⋊D4 | C23.23D10 | C23.18D10 | Dic5⋊D4 | D4×C20 | C4×D4 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{118}D_{10} % in TeX
G:=Group("C4^2.118D10"); // GroupNames label
G:=SmallGroup(320,1236);
// by ID
G=gap.SmallGroup(320,1236);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀