metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.113D10, C10.192+ (1+4), (C4×D4)⋊20D5, (D4×C20)⋊22C2, (C4×D20)⋊32C2, C20⋊2D4⋊9C2, D10⋊D4⋊9C2, C4⋊C4.318D10, (C2×D4).219D10, C4.65(C4○D20), C20.6Q8⋊16C2, (C22×C4).47D10, D10.52(C4○D4), C20.110(C4○D4), (C4×C20).157C22, (C2×C20).700C23, (C2×C10).102C24, C22⋊C4.115D10, C23.D10⋊8C2, C2.20(D4⋊6D10), C23.99(C22×D5), (C2×D20).221C22, (D4×C10).262C22, C4⋊Dic5.200C22, (C2×Dic5).43C23, (C22×D5).36C23, C22.127(C23×D5), C23.D5.14C22, D10⋊C4.86C22, C23.23D10⋊17C2, (C22×C20).364C22, (C22×C10).172C23, C5⋊4(C22.47C24), (C4×Dic5).225C22, C10.D4.100C22, (D5×C4⋊C4)⋊16C2, (C4×C5⋊D4)⋊44C2, C2.25(D5×C4○D4), C4⋊C4⋊7D5⋊15C2, C2.51(C2×C4○D20), (C2×C4×D5).75C22, C10.142(C2×C4○D4), (C5×C4⋊C4).331C22, (C2×C4).285(C22×D5), (C2×C5⋊D4).17C22, (C5×C22⋊C4).126C22, SmallGroup(320,1230)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 838 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D5 [×3], C10 [×3], C10 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], Dic5 [×6], C20 [×2], C20 [×4], D10 [×2], D10 [×5], C2×C10, C2×C10 [×6], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D5 [×6], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.47C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×3], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20.6Q8, C4×D20, C23.D10 [×2], D10⋊D4 [×2], D5×C4⋊C4, C4⋊C4⋊7D5, C4×C5⋊D4 [×2], C23.23D10 [×2], C20⋊2D4 [×2], D4×C20, C42.113D10
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], C22.47C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊6D10, D5×C4○D4, C42.113D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 57 111 141)(2 142 112 58)(3 59 113 143)(4 144 114 60)(5 41 115 145)(6 146 116 42)(7 43 117 147)(8 148 118 44)(9 45 119 149)(10 150 120 46)(11 47 101 151)(12 152 102 48)(13 49 103 153)(14 154 104 50)(15 51 105 155)(16 156 106 52)(17 53 107 157)(18 158 108 54)(19 55 109 159)(20 160 110 56)(21 135 68 90)(22 91 69 136)(23 137 70 92)(24 93 71 138)(25 139 72 94)(26 95 73 140)(27 121 74 96)(28 97 75 122)(29 123 76 98)(30 99 77 124)(31 125 78 100)(32 81 79 126)(33 127 80 82)(34 83 61 128)(35 129 62 84)(36 85 63 130)(37 131 64 86)(38 87 65 132)(39 133 66 88)(40 89 67 134)
(1 140 11 130)(2 121 12 131)(3 122 13 132)(4 123 14 133)(5 124 15 134)(6 125 16 135)(7 126 17 136)(8 127 18 137)(9 128 19 138)(10 129 20 139)(21 42 31 52)(22 43 32 53)(23 44 33 54)(24 45 34 55)(25 46 35 56)(26 47 36 57)(27 48 37 58)(28 49 38 59)(29 50 39 60)(30 51 40 41)(61 159 71 149)(62 160 72 150)(63 141 73 151)(64 142 74 152)(65 143 75 153)(66 144 76 154)(67 145 77 155)(68 146 78 156)(69 147 79 157)(70 148 80 158)(81 107 91 117)(82 108 92 118)(83 109 93 119)(84 110 94 120)(85 111 95 101)(86 112 96 102)(87 113 97 103)(88 114 98 104)(89 115 99 105)(90 116 100 106)
(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 72 11 62)(2 61 12 71)(3 70 13 80)(4 79 14 69)(5 68 15 78)(6 77 16 67)(7 66 17 76)(8 75 18 65)(9 64 19 74)(10 73 20 63)(21 105 31 115)(22 114 32 104)(23 103 33 113)(24 112 34 102)(25 101 35 111)(26 110 36 120)(27 119 37 109)(28 108 38 118)(29 117 39 107)(30 106 40 116)(41 135 51 125)(42 124 52 134)(43 133 53 123)(44 122 54 132)(45 131 55 121)(46 140 56 130)(47 129 57 139)(48 138 58 128)(49 127 59 137)(50 136 60 126)(81 154 91 144)(82 143 92 153)(83 152 93 142)(84 141 94 151)(85 150 95 160)(86 159 96 149)(87 148 97 158)(88 157 98 147)(89 146 99 156)(90 155 100 145)
G:=sub<Sym(160)| (1,57,111,141)(2,142,112,58)(3,59,113,143)(4,144,114,60)(5,41,115,145)(6,146,116,42)(7,43,117,147)(8,148,118,44)(9,45,119,149)(10,150,120,46)(11,47,101,151)(12,152,102,48)(13,49,103,153)(14,154,104,50)(15,51,105,155)(16,156,106,52)(17,53,107,157)(18,158,108,54)(19,55,109,159)(20,160,110,56)(21,135,68,90)(22,91,69,136)(23,137,70,92)(24,93,71,138)(25,139,72,94)(26,95,73,140)(27,121,74,96)(28,97,75,122)(29,123,76,98)(30,99,77,124)(31,125,78,100)(32,81,79,126)(33,127,80,82)(34,83,61,128)(35,129,62,84)(36,85,63,130)(37,131,64,86)(38,87,65,132)(39,133,66,88)(40,89,67,134), (1,140,11,130)(2,121,12,131)(3,122,13,132)(4,123,14,133)(5,124,15,134)(6,125,16,135)(7,126,17,136)(8,127,18,137)(9,128,19,138)(10,129,20,139)(21,42,31,52)(22,43,32,53)(23,44,33,54)(24,45,34,55)(25,46,35,56)(26,47,36,57)(27,48,37,58)(28,49,38,59)(29,50,39,60)(30,51,40,41)(61,159,71,149)(62,160,72,150)(63,141,73,151)(64,142,74,152)(65,143,75,153)(66,144,76,154)(67,145,77,155)(68,146,78,156)(69,147,79,157)(70,148,80,158)(81,107,91,117)(82,108,92,118)(83,109,93,119)(84,110,94,120)(85,111,95,101)(86,112,96,102)(87,113,97,103)(88,114,98,104)(89,115,99,105)(90,116,100,106), (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), (1,72,11,62)(2,61,12,71)(3,70,13,80)(4,79,14,69)(5,68,15,78)(6,77,16,67)(7,66,17,76)(8,75,18,65)(9,64,19,74)(10,73,20,63)(21,105,31,115)(22,114,32,104)(23,103,33,113)(24,112,34,102)(25,101,35,111)(26,110,36,120)(27,119,37,109)(28,108,38,118)(29,117,39,107)(30,106,40,116)(41,135,51,125)(42,124,52,134)(43,133,53,123)(44,122,54,132)(45,131,55,121)(46,140,56,130)(47,129,57,139)(48,138,58,128)(49,127,59,137)(50,136,60,126)(81,154,91,144)(82,143,92,153)(83,152,93,142)(84,141,94,151)(85,150,95,160)(86,159,96,149)(87,148,97,158)(88,157,98,147)(89,146,99,156)(90,155,100,145)>;
G:=Group( (1,57,111,141)(2,142,112,58)(3,59,113,143)(4,144,114,60)(5,41,115,145)(6,146,116,42)(7,43,117,147)(8,148,118,44)(9,45,119,149)(10,150,120,46)(11,47,101,151)(12,152,102,48)(13,49,103,153)(14,154,104,50)(15,51,105,155)(16,156,106,52)(17,53,107,157)(18,158,108,54)(19,55,109,159)(20,160,110,56)(21,135,68,90)(22,91,69,136)(23,137,70,92)(24,93,71,138)(25,139,72,94)(26,95,73,140)(27,121,74,96)(28,97,75,122)(29,123,76,98)(30,99,77,124)(31,125,78,100)(32,81,79,126)(33,127,80,82)(34,83,61,128)(35,129,62,84)(36,85,63,130)(37,131,64,86)(38,87,65,132)(39,133,66,88)(40,89,67,134), (1,140,11,130)(2,121,12,131)(3,122,13,132)(4,123,14,133)(5,124,15,134)(6,125,16,135)(7,126,17,136)(8,127,18,137)(9,128,19,138)(10,129,20,139)(21,42,31,52)(22,43,32,53)(23,44,33,54)(24,45,34,55)(25,46,35,56)(26,47,36,57)(27,48,37,58)(28,49,38,59)(29,50,39,60)(30,51,40,41)(61,159,71,149)(62,160,72,150)(63,141,73,151)(64,142,74,152)(65,143,75,153)(66,144,76,154)(67,145,77,155)(68,146,78,156)(69,147,79,157)(70,148,80,158)(81,107,91,117)(82,108,92,118)(83,109,93,119)(84,110,94,120)(85,111,95,101)(86,112,96,102)(87,113,97,103)(88,114,98,104)(89,115,99,105)(90,116,100,106), (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), (1,72,11,62)(2,61,12,71)(3,70,13,80)(4,79,14,69)(5,68,15,78)(6,77,16,67)(7,66,17,76)(8,75,18,65)(9,64,19,74)(10,73,20,63)(21,105,31,115)(22,114,32,104)(23,103,33,113)(24,112,34,102)(25,101,35,111)(26,110,36,120)(27,119,37,109)(28,108,38,118)(29,117,39,107)(30,106,40,116)(41,135,51,125)(42,124,52,134)(43,133,53,123)(44,122,54,132)(45,131,55,121)(46,140,56,130)(47,129,57,139)(48,138,58,128)(49,127,59,137)(50,136,60,126)(81,154,91,144)(82,143,92,153)(83,152,93,142)(84,141,94,151)(85,150,95,160)(86,159,96,149)(87,148,97,158)(88,157,98,147)(89,146,99,156)(90,155,100,145) );
G=PermutationGroup([(1,57,111,141),(2,142,112,58),(3,59,113,143),(4,144,114,60),(5,41,115,145),(6,146,116,42),(7,43,117,147),(8,148,118,44),(9,45,119,149),(10,150,120,46),(11,47,101,151),(12,152,102,48),(13,49,103,153),(14,154,104,50),(15,51,105,155),(16,156,106,52),(17,53,107,157),(18,158,108,54),(19,55,109,159),(20,160,110,56),(21,135,68,90),(22,91,69,136),(23,137,70,92),(24,93,71,138),(25,139,72,94),(26,95,73,140),(27,121,74,96),(28,97,75,122),(29,123,76,98),(30,99,77,124),(31,125,78,100),(32,81,79,126),(33,127,80,82),(34,83,61,128),(35,129,62,84),(36,85,63,130),(37,131,64,86),(38,87,65,132),(39,133,66,88),(40,89,67,134)], [(1,140,11,130),(2,121,12,131),(3,122,13,132),(4,123,14,133),(5,124,15,134),(6,125,16,135),(7,126,17,136),(8,127,18,137),(9,128,19,138),(10,129,20,139),(21,42,31,52),(22,43,32,53),(23,44,33,54),(24,45,34,55),(25,46,35,56),(26,47,36,57),(27,48,37,58),(28,49,38,59),(29,50,39,60),(30,51,40,41),(61,159,71,149),(62,160,72,150),(63,141,73,151),(64,142,74,152),(65,143,75,153),(66,144,76,154),(67,145,77,155),(68,146,78,156),(69,147,79,157),(70,148,80,158),(81,107,91,117),(82,108,92,118),(83,109,93,119),(84,110,94,120),(85,111,95,101),(86,112,96,102),(87,113,97,103),(88,114,98,104),(89,115,99,105),(90,116,100,106)], [(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)], [(1,72,11,62),(2,61,12,71),(3,70,13,80),(4,79,14,69),(5,68,15,78),(6,77,16,67),(7,66,17,76),(8,75,18,65),(9,64,19,74),(10,73,20,63),(21,105,31,115),(22,114,32,104),(23,103,33,113),(24,112,34,102),(25,101,35,111),(26,110,36,120),(27,119,37,109),(28,108,38,118),(29,117,39,107),(30,106,40,116),(41,135,51,125),(42,124,52,134),(43,133,53,123),(44,122,54,132),(45,131,55,121),(46,140,56,130),(47,129,57,139),(48,138,58,128),(49,127,59,137),(50,136,60,126),(81,154,91,144),(82,143,92,153),(83,152,93,142),(84,141,94,151),(85,150,95,160),(86,159,96,149),(87,148,97,158),(88,157,98,147),(89,146,99,156),(90,155,100,145)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 28 | 28 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
| 21 | 20 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 32 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,9,0,0,0,0,9],[28,13,0,0,28,32,0,0,0,0,0,9,0,0,9,0],[21,23,0,0,20,20,0,0,0,0,0,32,0,0,32,0] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 2 | ··· | 2 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 |
| kernel | C42.113D10 | C20.6Q8 | C4×D20 | C23.D10 | D10⋊D4 | D5×C4⋊C4 | C4⋊C4⋊7D5 | C4×C5⋊D4 | C23.23D10 | C20⋊2D4 | D4×C20 | C4×D4 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{113}D_{10} % in TeX
G:=Group("C4^2.113D10"); // GroupNames label
G:=SmallGroup(320,1230);
// by ID
G=gap.SmallGroup(320,1230);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀