metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.94D10, C10.512- (1+4), C4⋊C4.271D10, C42⋊D5⋊2C2, C20.6Q8⋊6C2, D10⋊2Q8⋊12C2, C4.95(C4○D20), C42⋊C2⋊13D5, (C2×C10).73C24, (C4×C20).24C22, C22⋊C4.97D10, C4.Dic10⋊12C2, Dic5⋊3Q8⋊12C2, D10.30(C4○D4), C20.197(C4○D4), C20.48D4⋊29C2, (C2×C20).148C23, (C22×C4).194D10, C23.D10⋊3C2, C4⋊Dic5.34C22, C23.85(C22×D5), D10.12D4.1C2, (C2×Dic5).26C23, (C4×Dic5).77C22, C22.102(C23×D5), C2.9(D4.10D10), C23.D5.96C22, D10⋊C4.96C22, (C22×C10).143C23, (C22×C20).231C22, C5⋊2(C22.46C24), C10.D4.97C22, (C22×D5).176C23, (C2×Dic10).149C22, (D5×C4⋊C4)⋊12C2, C2.12(D5×C4○D4), (C4×C5⋊D4).5C2, C2.32(C2×C4○D20), C10.30(C2×C4○D4), (C2×C4×D5).69C22, (C5×C42⋊C2)⋊15C2, (C5×C4⋊C4).309C22, (C2×C4).275(C22×D5), (C2×C5⋊D4).111C22, (C5×C22⋊C4).113C22, SmallGroup(320,1201)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 662 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic5 [×7], C20 [×2], C20 [×5], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2, C42⋊C2 [×2], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], Dic10 [×2], C4×D5 [×6], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×C10, C22.46C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×8], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×C5⋊D4, C22×C20, C20.6Q8 [×2], C42⋊D5 [×2], C23.D10 [×2], D10.12D4 [×2], Dic5⋊3Q8, C4.Dic10, D5×C4⋊C4, D10⋊2Q8, C20.48D4, C4×C5⋊D4, C5×C42⋊C2, C42.94D10
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], C22.46C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.94D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 63 42 109)(2 74 43 120)(3 65 44 111)(4 76 45 102)(5 67 46 113)(6 78 47 104)(7 69 48 115)(8 80 49 106)(9 71 50 117)(10 62 51 108)(11 73 52 119)(12 64 53 110)(13 75 54 101)(14 66 55 112)(15 77 56 103)(16 68 57 114)(17 79 58 105)(18 70 59 116)(19 61 60 107)(20 72 41 118)(21 90 139 148)(22 81 140 159)(23 92 121 150)(24 83 122 141)(25 94 123 152)(26 85 124 143)(27 96 125 154)(28 87 126 145)(29 98 127 156)(30 89 128 147)(31 100 129 158)(32 91 130 149)(33 82 131 160)(34 93 132 151)(35 84 133 142)(36 95 134 153)(37 86 135 144)(38 97 136 155)(39 88 137 146)(40 99 138 157)
(1 139 11 129)(2 140 12 130)(3 121 13 131)(4 122 14 132)(5 123 15 133)(6 124 16 134)(7 125 17 135)(8 126 18 136)(9 127 19 137)(10 128 20 138)(21 52 31 42)(22 53 32 43)(23 54 33 44)(24 55 34 45)(25 56 35 46)(26 57 36 47)(27 58 37 48)(28 59 38 49)(29 60 39 50)(30 41 40 51)(61 146 71 156)(62 147 72 157)(63 148 73 158)(64 149 74 159)(65 150 75 160)(66 151 76 141)(67 152 77 142)(68 153 78 143)(69 154 79 144)(70 155 80 145)(81 110 91 120)(82 111 92 101)(83 112 93 102)(84 113 94 103)(85 114 95 104)(86 115 96 105)(87 116 97 106)(88 117 98 107)(89 118 99 108)(90 119 100 109)
(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 20 11 10)(2 9 12 19)(3 18 13 8)(4 7 14 17)(5 16 15 6)(21 138 31 128)(22 127 32 137)(23 136 33 126)(24 125 34 135)(25 134 35 124)(26 123 36 133)(27 132 37 122)(28 121 38 131)(29 130 39 140)(30 139 40 129)(41 52 51 42)(43 50 53 60)(44 59 54 49)(45 48 55 58)(46 57 56 47)(61 64 71 74)(62 73 72 63)(65 80 75 70)(66 69 76 79)(67 78 77 68)(81 146 91 156)(82 155 92 145)(83 144 93 154)(84 153 94 143)(85 142 95 152)(86 151 96 141)(87 160 97 150)(88 149 98 159)(89 158 99 148)(90 147 100 157)(101 116 111 106)(102 105 112 115)(103 114 113 104)(107 110 117 120)(108 119 118 109)
G:=sub<Sym(160)| (1,63,42,109)(2,74,43,120)(3,65,44,111)(4,76,45,102)(5,67,46,113)(6,78,47,104)(7,69,48,115)(8,80,49,106)(9,71,50,117)(10,62,51,108)(11,73,52,119)(12,64,53,110)(13,75,54,101)(14,66,55,112)(15,77,56,103)(16,68,57,114)(17,79,58,105)(18,70,59,116)(19,61,60,107)(20,72,41,118)(21,90,139,148)(22,81,140,159)(23,92,121,150)(24,83,122,141)(25,94,123,152)(26,85,124,143)(27,96,125,154)(28,87,126,145)(29,98,127,156)(30,89,128,147)(31,100,129,158)(32,91,130,149)(33,82,131,160)(34,93,132,151)(35,84,133,142)(36,95,134,153)(37,86,135,144)(38,97,136,155)(39,88,137,146)(40,99,138,157), (1,139,11,129)(2,140,12,130)(3,121,13,131)(4,122,14,132)(5,123,15,133)(6,124,16,134)(7,125,17,135)(8,126,18,136)(9,127,19,137)(10,128,20,138)(21,52,31,42)(22,53,32,43)(23,54,33,44)(24,55,34,45)(25,56,35,46)(26,57,36,47)(27,58,37,48)(28,59,38,49)(29,60,39,50)(30,41,40,51)(61,146,71,156)(62,147,72,157)(63,148,73,158)(64,149,74,159)(65,150,75,160)(66,151,76,141)(67,152,77,142)(68,153,78,143)(69,154,79,144)(70,155,80,145)(81,110,91,120)(82,111,92,101)(83,112,93,102)(84,113,94,103)(85,114,95,104)(86,115,96,105)(87,116,97,106)(88,117,98,107)(89,118,99,108)(90,119,100,109), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,138,31,128)(22,127,32,137)(23,136,33,126)(24,125,34,135)(25,134,35,124)(26,123,36,133)(27,132,37,122)(28,121,38,131)(29,130,39,140)(30,139,40,129)(41,52,51,42)(43,50,53,60)(44,59,54,49)(45,48,55,58)(46,57,56,47)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)(81,146,91,156)(82,155,92,145)(83,144,93,154)(84,153,94,143)(85,142,95,152)(86,151,96,141)(87,160,97,150)(88,149,98,159)(89,158,99,148)(90,147,100,157)(101,116,111,106)(102,105,112,115)(103,114,113,104)(107,110,117,120)(108,119,118,109)>;
G:=Group( (1,63,42,109)(2,74,43,120)(3,65,44,111)(4,76,45,102)(5,67,46,113)(6,78,47,104)(7,69,48,115)(8,80,49,106)(9,71,50,117)(10,62,51,108)(11,73,52,119)(12,64,53,110)(13,75,54,101)(14,66,55,112)(15,77,56,103)(16,68,57,114)(17,79,58,105)(18,70,59,116)(19,61,60,107)(20,72,41,118)(21,90,139,148)(22,81,140,159)(23,92,121,150)(24,83,122,141)(25,94,123,152)(26,85,124,143)(27,96,125,154)(28,87,126,145)(29,98,127,156)(30,89,128,147)(31,100,129,158)(32,91,130,149)(33,82,131,160)(34,93,132,151)(35,84,133,142)(36,95,134,153)(37,86,135,144)(38,97,136,155)(39,88,137,146)(40,99,138,157), (1,139,11,129)(2,140,12,130)(3,121,13,131)(4,122,14,132)(5,123,15,133)(6,124,16,134)(7,125,17,135)(8,126,18,136)(9,127,19,137)(10,128,20,138)(21,52,31,42)(22,53,32,43)(23,54,33,44)(24,55,34,45)(25,56,35,46)(26,57,36,47)(27,58,37,48)(28,59,38,49)(29,60,39,50)(30,41,40,51)(61,146,71,156)(62,147,72,157)(63,148,73,158)(64,149,74,159)(65,150,75,160)(66,151,76,141)(67,152,77,142)(68,153,78,143)(69,154,79,144)(70,155,80,145)(81,110,91,120)(82,111,92,101)(83,112,93,102)(84,113,94,103)(85,114,95,104)(86,115,96,105)(87,116,97,106)(88,117,98,107)(89,118,99,108)(90,119,100,109), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,138,31,128)(22,127,32,137)(23,136,33,126)(24,125,34,135)(25,134,35,124)(26,123,36,133)(27,132,37,122)(28,121,38,131)(29,130,39,140)(30,139,40,129)(41,52,51,42)(43,50,53,60)(44,59,54,49)(45,48,55,58)(46,57,56,47)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)(81,146,91,156)(82,155,92,145)(83,144,93,154)(84,153,94,143)(85,142,95,152)(86,151,96,141)(87,160,97,150)(88,149,98,159)(89,158,99,148)(90,147,100,157)(101,116,111,106)(102,105,112,115)(103,114,113,104)(107,110,117,120)(108,119,118,109) );
G=PermutationGroup([(1,63,42,109),(2,74,43,120),(3,65,44,111),(4,76,45,102),(5,67,46,113),(6,78,47,104),(7,69,48,115),(8,80,49,106),(9,71,50,117),(10,62,51,108),(11,73,52,119),(12,64,53,110),(13,75,54,101),(14,66,55,112),(15,77,56,103),(16,68,57,114),(17,79,58,105),(18,70,59,116),(19,61,60,107),(20,72,41,118),(21,90,139,148),(22,81,140,159),(23,92,121,150),(24,83,122,141),(25,94,123,152),(26,85,124,143),(27,96,125,154),(28,87,126,145),(29,98,127,156),(30,89,128,147),(31,100,129,158),(32,91,130,149),(33,82,131,160),(34,93,132,151),(35,84,133,142),(36,95,134,153),(37,86,135,144),(38,97,136,155),(39,88,137,146),(40,99,138,157)], [(1,139,11,129),(2,140,12,130),(3,121,13,131),(4,122,14,132),(5,123,15,133),(6,124,16,134),(7,125,17,135),(8,126,18,136),(9,127,19,137),(10,128,20,138),(21,52,31,42),(22,53,32,43),(23,54,33,44),(24,55,34,45),(25,56,35,46),(26,57,36,47),(27,58,37,48),(28,59,38,49),(29,60,39,50),(30,41,40,51),(61,146,71,156),(62,147,72,157),(63,148,73,158),(64,149,74,159),(65,150,75,160),(66,151,76,141),(67,152,77,142),(68,153,78,143),(69,154,79,144),(70,155,80,145),(81,110,91,120),(82,111,92,101),(83,112,93,102),(84,113,94,103),(85,114,95,104),(86,115,96,105),(87,116,97,106),(88,117,98,107),(89,118,99,108),(90,119,100,109)], [(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)], [(1,20,11,10),(2,9,12,19),(3,18,13,8),(4,7,14,17),(5,16,15,6),(21,138,31,128),(22,127,32,137),(23,136,33,126),(24,125,34,135),(25,134,35,124),(26,123,36,133),(27,132,37,122),(28,121,38,131),(29,130,39,140),(30,139,40,129),(41,52,51,42),(43,50,53,60),(44,59,54,49),(45,48,55,58),(46,57,56,47),(61,64,71,74),(62,73,72,63),(65,80,75,70),(66,69,76,79),(67,78,77,68),(81,146,91,156),(82,155,92,145),(83,144,93,154),(84,153,94,143),(85,142,95,152),(86,151,96,141),(87,160,97,150),(88,149,98,159),(89,158,99,148),(90,147,100,157),(101,116,111,106),(102,105,112,115),(103,114,113,104),(107,110,117,120),(108,119,118,109)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 34 | 39 | 0 | 0 |
| 24 | 7 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 23 | 35 |
| 0 | 0 | 6 | 18 |
| 13 | 35 | 0 | 0 |
| 1 | 28 | 0 | 0 |
| 0 | 0 | 40 | 35 |
| 0 | 0 | 6 | 35 |
| 28 | 6 | 0 | 0 |
| 40 | 13 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 6 | 1 |
G:=sub<GL(4,GF(41))| [34,24,0,0,39,7,0,0,0,0,9,0,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,23,6,0,0,35,18],[13,1,0,0,35,28,0,0,0,0,40,6,0,0,35,35],[28,40,0,0,6,13,0,0,0,0,40,6,0,0,0,1] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | D5×C4○D4 | D4.10D10 |
| kernel | C42.94D10 | C20.6Q8 | C42⋊D5 | C23.D10 | D10.12D4 | Dic5⋊3Q8 | C4.Dic10 | D5×C4⋊C4 | D10⋊2Q8 | C20.48D4 | C4×C5⋊D4 | C5×C42⋊C2 | C42⋊C2 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{94}D_{10} % in TeX
G:=Group("C4^2.94D10"); // GroupNames label
G:=SmallGroup(320,1201);
// by ID
G=gap.SmallGroup(320,1201);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,675,136,12550]);
// 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*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀