metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.241D10, C4⋊Q8⋊20D5, (C4×D5)⋊5Q8, C4.40(Q8×D5), D10.5(C2×Q8), C20.54(C2×Q8), C4⋊C4.219D10, C20⋊2Q8⋊36C2, (Q8×Dic5)⋊22C2, (C2×Q8).147D10, C4.Dic10⋊42C2, Dic5.34(C2×Q8), (D5×C42).10C2, Dic5⋊3Q8⋊42C2, C20.136(C4○D4), C4.41(D4⋊2D5), C10.48(C22×Q8), (C2×C10).272C24, (C4×C20).213C22, (C2×C20).105C23, D10⋊2Q8.14C2, D10⋊3Q8.12C2, C4.22(Q8⋊2D5), C4⋊Dic5.251C22, (Q8×C10).139C22, C22.293(C23×D5), D10⋊C4.51C22, C5⋊6(C23.37C23), (C2×Dic5).143C23, (C4×Dic5).169C22, C10.D4.61C22, (C22×D5).243C23, (C2×Dic10).196C22, C2.31(C2×Q8×D5), (C5×C4⋊Q8)⋊14C2, C4⋊C4⋊7D5.14C2, C10.100(C2×C4○D4), C2.64(C2×D4⋊2D5), C2.29(C2×Q8⋊2D5), (C2×C4×D5).322C22, (C5×C4⋊C4).215C22, (C2×C4).600(C22×D5), SmallGroup(320,1400)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 654 in 222 conjugacy classes, 111 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×12], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], Q8 [×8], C23, D5 [×2], C10 [×3], C42, C42 [×7], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×6], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8, C4⋊Q8, Dic10 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C22×D5, C23.37C23, C4×Dic5 [×3], C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×8], D10⋊C4 [×4], C4×C20, C5×C4⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5 [×3], Q8×C10 [×2], C20⋊2Q8, D5×C42, Dic5⋊3Q8 [×2], C4.Dic10 [×2], C4⋊C4⋊7D5 [×2], D10⋊2Q8 [×2], Q8×Dic5 [×2], D10⋊3Q8 [×2], C5×C4⋊Q8, C42.241D10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C22×Q8, C2×C4○D4 [×2], C22×D5 [×7], C23.37C23, D4⋊2D5 [×2], Q8×D5 [×2], Q8⋊2D5 [×2], C23×D5, C2×D4⋊2D5, C2×Q8×D5, C2×Q8⋊2D5, C42.241D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c9 >
►On 160 points
Generators in S160
(1 65 137 117)(2 118 138 66)(3 67 139 119)(4 120 140 68)(5 69 121 101)(6 102 122 70)(7 71 123 103)(8 104 124 72)(9 73 125 105)(10 106 126 74)(11 75 127 107)(12 108 128 76)(13 77 129 109)(14 110 130 78)(15 79 131 111)(16 112 132 80)(17 61 133 113)(18 114 134 62)(19 63 135 115)(20 116 136 64)(21 152 41 82)(22 83 42 153)(23 154 43 84)(24 85 44 155)(25 156 45 86)(26 87 46 157)(27 158 47 88)(28 89 48 159)(29 160 49 90)(30 91 50 141)(31 142 51 92)(32 93 52 143)(33 144 53 94)(34 95 54 145)(35 146 55 96)(36 97 56 147)(37 148 57 98)(38 99 58 149)(39 150 59 100)(40 81 60 151)
(1 83 11 93)(2 94 12 84)(3 85 13 95)(4 96 14 86)(5 87 15 97)(6 98 16 88)(7 89 17 99)(8 100 18 90)(9 91 19 81)(10 82 20 92)(21 116 31 106)(22 107 32 117)(23 118 33 108)(24 109 34 119)(25 120 35 110)(26 111 36 101)(27 102 37 112)(28 113 38 103)(29 104 39 114)(30 115 40 105)(41 64 51 74)(42 75 52 65)(43 66 53 76)(44 77 54 67)(45 68 55 78)(46 79 56 69)(47 70 57 80)(48 61 58 71)(49 72 59 62)(50 63 60 73)(121 157 131 147)(122 148 132 158)(123 159 133 149)(124 150 134 160)(125 141 135 151)(126 152 136 142)(127 143 137 153)(128 154 138 144)(129 145 139 155)(130 156 140 146)
(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 10 127 136)(2 135 128 9)(3 8 129 134)(4 133 130 7)(5 6 131 132)(11 20 137 126)(12 125 138 19)(13 18 139 124)(14 123 140 17)(15 16 121 122)(21 42 51 32)(22 31 52 41)(23 60 53 30)(24 29 54 59)(25 58 55 28)(26 27 56 57)(33 50 43 40)(34 39 44 49)(35 48 45 38)(36 37 46 47)(61 78 103 120)(62 119 104 77)(63 76 105 118)(64 117 106 75)(65 74 107 116)(66 115 108 73)(67 72 109 114)(68 113 110 71)(69 70 111 112)(79 80 101 102)(81 94 141 154)(82 153 142 93)(83 92 143 152)(84 151 144 91)(85 90 145 150)(86 149 146 89)(87 88 147 148)(95 100 155 160)(96 159 156 99)(97 98 157 158)
G:=sub<Sym(160)| (1,65,137,117)(2,118,138,66)(3,67,139,119)(4,120,140,68)(5,69,121,101)(6,102,122,70)(7,71,123,103)(8,104,124,72)(9,73,125,105)(10,106,126,74)(11,75,127,107)(12,108,128,76)(13,77,129,109)(14,110,130,78)(15,79,131,111)(16,112,132,80)(17,61,133,113)(18,114,134,62)(19,63,135,115)(20,116,136,64)(21,152,41,82)(22,83,42,153)(23,154,43,84)(24,85,44,155)(25,156,45,86)(26,87,46,157)(27,158,47,88)(28,89,48,159)(29,160,49,90)(30,91,50,141)(31,142,51,92)(32,93,52,143)(33,144,53,94)(34,95,54,145)(35,146,55,96)(36,97,56,147)(37,148,57,98)(38,99,58,149)(39,150,59,100)(40,81,60,151), (1,83,11,93)(2,94,12,84)(3,85,13,95)(4,96,14,86)(5,87,15,97)(6,98,16,88)(7,89,17,99)(8,100,18,90)(9,91,19,81)(10,82,20,92)(21,116,31,106)(22,107,32,117)(23,118,33,108)(24,109,34,119)(25,120,35,110)(26,111,36,101)(27,102,37,112)(28,113,38,103)(29,104,39,114)(30,115,40,105)(41,64,51,74)(42,75,52,65)(43,66,53,76)(44,77,54,67)(45,68,55,78)(46,79,56,69)(47,70,57,80)(48,61,58,71)(49,72,59,62)(50,63,60,73)(121,157,131,147)(122,148,132,158)(123,159,133,149)(124,150,134,160)(125,141,135,151)(126,152,136,142)(127,143,137,153)(128,154,138,144)(129,145,139,155)(130,156,140,146), (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), (1,10,127,136)(2,135,128,9)(3,8,129,134)(4,133,130,7)(5,6,131,132)(11,20,137,126)(12,125,138,19)(13,18,139,124)(14,123,140,17)(15,16,121,122)(21,42,51,32)(22,31,52,41)(23,60,53,30)(24,29,54,59)(25,58,55,28)(26,27,56,57)(33,50,43,40)(34,39,44,49)(35,48,45,38)(36,37,46,47)(61,78,103,120)(62,119,104,77)(63,76,105,118)(64,117,106,75)(65,74,107,116)(66,115,108,73)(67,72,109,114)(68,113,110,71)(69,70,111,112)(79,80,101,102)(81,94,141,154)(82,153,142,93)(83,92,143,152)(84,151,144,91)(85,90,145,150)(86,149,146,89)(87,88,147,148)(95,100,155,160)(96,159,156,99)(97,98,157,158)>;
G:=Group( (1,65,137,117)(2,118,138,66)(3,67,139,119)(4,120,140,68)(5,69,121,101)(6,102,122,70)(7,71,123,103)(8,104,124,72)(9,73,125,105)(10,106,126,74)(11,75,127,107)(12,108,128,76)(13,77,129,109)(14,110,130,78)(15,79,131,111)(16,112,132,80)(17,61,133,113)(18,114,134,62)(19,63,135,115)(20,116,136,64)(21,152,41,82)(22,83,42,153)(23,154,43,84)(24,85,44,155)(25,156,45,86)(26,87,46,157)(27,158,47,88)(28,89,48,159)(29,160,49,90)(30,91,50,141)(31,142,51,92)(32,93,52,143)(33,144,53,94)(34,95,54,145)(35,146,55,96)(36,97,56,147)(37,148,57,98)(38,99,58,149)(39,150,59,100)(40,81,60,151), (1,83,11,93)(2,94,12,84)(3,85,13,95)(4,96,14,86)(5,87,15,97)(6,98,16,88)(7,89,17,99)(8,100,18,90)(9,91,19,81)(10,82,20,92)(21,116,31,106)(22,107,32,117)(23,118,33,108)(24,109,34,119)(25,120,35,110)(26,111,36,101)(27,102,37,112)(28,113,38,103)(29,104,39,114)(30,115,40,105)(41,64,51,74)(42,75,52,65)(43,66,53,76)(44,77,54,67)(45,68,55,78)(46,79,56,69)(47,70,57,80)(48,61,58,71)(49,72,59,62)(50,63,60,73)(121,157,131,147)(122,148,132,158)(123,159,133,149)(124,150,134,160)(125,141,135,151)(126,152,136,142)(127,143,137,153)(128,154,138,144)(129,145,139,155)(130,156,140,146), (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), (1,10,127,136)(2,135,128,9)(3,8,129,134)(4,133,130,7)(5,6,131,132)(11,20,137,126)(12,125,138,19)(13,18,139,124)(14,123,140,17)(15,16,121,122)(21,42,51,32)(22,31,52,41)(23,60,53,30)(24,29,54,59)(25,58,55,28)(26,27,56,57)(33,50,43,40)(34,39,44,49)(35,48,45,38)(36,37,46,47)(61,78,103,120)(62,119,104,77)(63,76,105,118)(64,117,106,75)(65,74,107,116)(66,115,108,73)(67,72,109,114)(68,113,110,71)(69,70,111,112)(79,80,101,102)(81,94,141,154)(82,153,142,93)(83,92,143,152)(84,151,144,91)(85,90,145,150)(86,149,146,89)(87,88,147,148)(95,100,155,160)(96,159,156,99)(97,98,157,158) );
G=PermutationGroup([(1,65,137,117),(2,118,138,66),(3,67,139,119),(4,120,140,68),(5,69,121,101),(6,102,122,70),(7,71,123,103),(8,104,124,72),(9,73,125,105),(10,106,126,74),(11,75,127,107),(12,108,128,76),(13,77,129,109),(14,110,130,78),(15,79,131,111),(16,112,132,80),(17,61,133,113),(18,114,134,62),(19,63,135,115),(20,116,136,64),(21,152,41,82),(22,83,42,153),(23,154,43,84),(24,85,44,155),(25,156,45,86),(26,87,46,157),(27,158,47,88),(28,89,48,159),(29,160,49,90),(30,91,50,141),(31,142,51,92),(32,93,52,143),(33,144,53,94),(34,95,54,145),(35,146,55,96),(36,97,56,147),(37,148,57,98),(38,99,58,149),(39,150,59,100),(40,81,60,151)], [(1,83,11,93),(2,94,12,84),(3,85,13,95),(4,96,14,86),(5,87,15,97),(6,98,16,88),(7,89,17,99),(8,100,18,90),(9,91,19,81),(10,82,20,92),(21,116,31,106),(22,107,32,117),(23,118,33,108),(24,109,34,119),(25,120,35,110),(26,111,36,101),(27,102,37,112),(28,113,38,103),(29,104,39,114),(30,115,40,105),(41,64,51,74),(42,75,52,65),(43,66,53,76),(44,77,54,67),(45,68,55,78),(46,79,56,69),(47,70,57,80),(48,61,58,71),(49,72,59,62),(50,63,60,73),(121,157,131,147),(122,148,132,158),(123,159,133,149),(124,150,134,160),(125,141,135,151),(126,152,136,142),(127,143,137,153),(128,154,138,144),(129,145,139,155),(130,156,140,146)], [(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)], [(1,10,127,136),(2,135,128,9),(3,8,129,134),(4,133,130,7),(5,6,131,132),(11,20,137,126),(12,125,138,19),(13,18,139,124),(14,123,140,17),(15,16,121,122),(21,42,51,32),(22,31,52,41),(23,60,53,30),(24,29,54,59),(25,58,55,28),(26,27,56,57),(33,50,43,40),(34,39,44,49),(35,48,45,38),(36,37,46,47),(61,78,103,120),(62,119,104,77),(63,76,105,118),(64,117,106,75),(65,74,107,116),(66,115,108,73),(67,72,109,114),(68,113,110,71),(69,70,111,112),(79,80,101,102),(81,94,141,154),(82,153,142,93),(83,92,143,152),(84,151,144,91),(85,90,145,150),(86,149,146,89),(87,88,147,148),(95,100,155,160),(96,159,156,99),(97,98,157,158)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | D10 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 |
| kernel | C42.241D10 | C20⋊2Q8 | D5×C42 | Dic5⋊3Q8 | C4.Dic10 | C4⋊C4⋊7D5 | D10⋊2Q8 | Q8×Dic5 | D10⋊3Q8 | C5×C4⋊Q8 | C4×D5 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 8 | 2 | 8 | 4 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{241}D_{10} % in TeX
G:=Group("C4^2.241D10"); // GroupNames label
G:=SmallGroup(320,1400);
// by ID
G=gap.SmallGroup(320,1400);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,570,185,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀