metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.134D10, C10.142- (1+4), (C4×Q8)⋊16D5, (Q8×C20)⋊18C2, C4⋊C4.301D10, (C4×Dic10)⋊40C2, C4.50(C4○D20), (C2×Q8).182D10, C20.6Q8⋊27C2, C4.Dic10⋊17C2, C42⋊2D5.2C2, C42⋊D5.5C2, Dic5⋊Q8⋊10C2, C20.121(C4○D4), (C4×C20).179C22, (C2×C10).127C24, (C2×C20).624C23, D10⋊3Q8.11C2, D10⋊2Q8.11C2, Dic5.Q8⋊10C2, C4⋊Dic5.370C22, (Q8×C10).227C22, (C4×Dic5).95C22, (C2×Dic5).58C23, (C22×D5).49C23, C22.148(C23×D5), C5⋊2(C22.35C24), C10.D4.78C22, C2.24(D4.10D10), D10⋊C4.126C22, C2.15(Q8.10D10), (C2×Dic10).300C22, C4⋊C4⋊D5.1C2, C10.57(C2×C4○D4), C2.66(C2×C4○D20), (C2×C4×D5).86C22, (C5×C4⋊C4).355C22, (C2×C4).290(C22×D5), SmallGroup(320,1255)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 574 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], Q8 [×4], C23, D5, C10 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C2×Q8, C2×Q8, Dic5 [×7], C20 [×2], C20 [×6], D10 [×3], C2×C10, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22.35C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×10], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×4], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, Q8×C10, C4×Dic10, C20.6Q8 [×2], C42⋊D5, C42⋊2D5 [×2], Dic5.Q8 [×2], C4.Dic10, D10⋊2Q8, C4⋊C4⋊D5 [×2], Dic5⋊Q8, D10⋊3Q8, Q8×C20, C42.134D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2- (1+4) [×2], C22×D5 [×7], C22.35C24, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D4.10D10, C42.134D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c9 >
►On 160 points
Generators in S160
(1 93 124 110)(2 111 125 94)(3 95 126 112)(4 113 127 96)(5 97 128 114)(6 115 129 98)(7 99 130 116)(8 117 131 100)(9 81 132 118)(10 119 133 82)(11 83 134 120)(12 101 135 84)(13 85 136 102)(14 103 137 86)(15 87 138 104)(16 105 139 88)(17 89 140 106)(18 107 121 90)(19 91 122 108)(20 109 123 92)(21 68 153 41)(22 42 154 69)(23 70 155 43)(24 44 156 71)(25 72 157 45)(26 46 158 73)(27 74 159 47)(28 48 160 75)(29 76 141 49)(30 50 142 77)(31 78 143 51)(32 52 144 79)(33 80 145 53)(34 54 146 61)(35 62 147 55)(36 56 148 63)(37 64 149 57)(38 58 150 65)(39 66 151 59)(40 60 152 67)
(1 36 134 158)(2 37 135 159)(3 38 136 160)(4 39 137 141)(5 40 138 142)(6 21 139 143)(7 22 140 144)(8 23 121 145)(9 24 122 146)(10 25 123 147)(11 26 124 148)(12 27 125 149)(13 28 126 150)(14 29 127 151)(15 30 128 152)(16 31 129 153)(17 32 130 154)(18 33 131 155)(19 34 132 156)(20 35 133 157)(41 105 78 98)(42 106 79 99)(43 107 80 100)(44 108 61 81)(45 109 62 82)(46 110 63 83)(47 111 64 84)(48 112 65 85)(49 113 66 86)(50 114 67 87)(51 115 68 88)(52 116 69 89)(53 117 70 90)(54 118 71 91)(55 119 72 92)(56 120 73 93)(57 101 74 94)(58 102 75 95)(59 103 76 96)(60 104 77 97)
(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 10 124 133)(2 132 125 9)(3 8 126 131)(4 130 127 7)(5 6 128 129)(11 20 134 123)(12 122 135 19)(13 18 136 121)(14 140 137 17)(15 16 138 139)(21 40 153 152)(22 151 154 39)(23 38 155 150)(24 149 156 37)(25 36 157 148)(26 147 158 35)(27 34 159 146)(28 145 160 33)(29 32 141 144)(30 143 142 31)(41 60 68 67)(42 66 69 59)(43 58 70 65)(44 64 71 57)(45 56 72 63)(46 62 73 55)(47 54 74 61)(48 80 75 53)(49 52 76 79)(50 78 77 51)(81 94 118 111)(82 110 119 93)(83 92 120 109)(84 108 101 91)(85 90 102 107)(86 106 103 89)(87 88 104 105)(95 100 112 117)(96 116 113 99)(97 98 114 115)
G:=sub<Sym(160)| (1,93,124,110)(2,111,125,94)(3,95,126,112)(4,113,127,96)(5,97,128,114)(6,115,129,98)(7,99,130,116)(8,117,131,100)(9,81,132,118)(10,119,133,82)(11,83,134,120)(12,101,135,84)(13,85,136,102)(14,103,137,86)(15,87,138,104)(16,105,139,88)(17,89,140,106)(18,107,121,90)(19,91,122,108)(20,109,123,92)(21,68,153,41)(22,42,154,69)(23,70,155,43)(24,44,156,71)(25,72,157,45)(26,46,158,73)(27,74,159,47)(28,48,160,75)(29,76,141,49)(30,50,142,77)(31,78,143,51)(32,52,144,79)(33,80,145,53)(34,54,146,61)(35,62,147,55)(36,56,148,63)(37,64,149,57)(38,58,150,65)(39,66,151,59)(40,60,152,67), (1,36,134,158)(2,37,135,159)(3,38,136,160)(4,39,137,141)(5,40,138,142)(6,21,139,143)(7,22,140,144)(8,23,121,145)(9,24,122,146)(10,25,123,147)(11,26,124,148)(12,27,125,149)(13,28,126,150)(14,29,127,151)(15,30,128,152)(16,31,129,153)(17,32,130,154)(18,33,131,155)(19,34,132,156)(20,35,133,157)(41,105,78,98)(42,106,79,99)(43,107,80,100)(44,108,61,81)(45,109,62,82)(46,110,63,83)(47,111,64,84)(48,112,65,85)(49,113,66,86)(50,114,67,87)(51,115,68,88)(52,116,69,89)(53,117,70,90)(54,118,71,91)(55,119,72,92)(56,120,73,93)(57,101,74,94)(58,102,75,95)(59,103,76,96)(60,104,77,97), (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), (1,10,124,133)(2,132,125,9)(3,8,126,131)(4,130,127,7)(5,6,128,129)(11,20,134,123)(12,122,135,19)(13,18,136,121)(14,140,137,17)(15,16,138,139)(21,40,153,152)(22,151,154,39)(23,38,155,150)(24,149,156,37)(25,36,157,148)(26,147,158,35)(27,34,159,146)(28,145,160,33)(29,32,141,144)(30,143,142,31)(41,60,68,67)(42,66,69,59)(43,58,70,65)(44,64,71,57)(45,56,72,63)(46,62,73,55)(47,54,74,61)(48,80,75,53)(49,52,76,79)(50,78,77,51)(81,94,118,111)(82,110,119,93)(83,92,120,109)(84,108,101,91)(85,90,102,107)(86,106,103,89)(87,88,104,105)(95,100,112,117)(96,116,113,99)(97,98,114,115)>;
G:=Group( (1,93,124,110)(2,111,125,94)(3,95,126,112)(4,113,127,96)(5,97,128,114)(6,115,129,98)(7,99,130,116)(8,117,131,100)(9,81,132,118)(10,119,133,82)(11,83,134,120)(12,101,135,84)(13,85,136,102)(14,103,137,86)(15,87,138,104)(16,105,139,88)(17,89,140,106)(18,107,121,90)(19,91,122,108)(20,109,123,92)(21,68,153,41)(22,42,154,69)(23,70,155,43)(24,44,156,71)(25,72,157,45)(26,46,158,73)(27,74,159,47)(28,48,160,75)(29,76,141,49)(30,50,142,77)(31,78,143,51)(32,52,144,79)(33,80,145,53)(34,54,146,61)(35,62,147,55)(36,56,148,63)(37,64,149,57)(38,58,150,65)(39,66,151,59)(40,60,152,67), (1,36,134,158)(2,37,135,159)(3,38,136,160)(4,39,137,141)(5,40,138,142)(6,21,139,143)(7,22,140,144)(8,23,121,145)(9,24,122,146)(10,25,123,147)(11,26,124,148)(12,27,125,149)(13,28,126,150)(14,29,127,151)(15,30,128,152)(16,31,129,153)(17,32,130,154)(18,33,131,155)(19,34,132,156)(20,35,133,157)(41,105,78,98)(42,106,79,99)(43,107,80,100)(44,108,61,81)(45,109,62,82)(46,110,63,83)(47,111,64,84)(48,112,65,85)(49,113,66,86)(50,114,67,87)(51,115,68,88)(52,116,69,89)(53,117,70,90)(54,118,71,91)(55,119,72,92)(56,120,73,93)(57,101,74,94)(58,102,75,95)(59,103,76,96)(60,104,77,97), (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), (1,10,124,133)(2,132,125,9)(3,8,126,131)(4,130,127,7)(5,6,128,129)(11,20,134,123)(12,122,135,19)(13,18,136,121)(14,140,137,17)(15,16,138,139)(21,40,153,152)(22,151,154,39)(23,38,155,150)(24,149,156,37)(25,36,157,148)(26,147,158,35)(27,34,159,146)(28,145,160,33)(29,32,141,144)(30,143,142,31)(41,60,68,67)(42,66,69,59)(43,58,70,65)(44,64,71,57)(45,56,72,63)(46,62,73,55)(47,54,74,61)(48,80,75,53)(49,52,76,79)(50,78,77,51)(81,94,118,111)(82,110,119,93)(83,92,120,109)(84,108,101,91)(85,90,102,107)(86,106,103,89)(87,88,104,105)(95,100,112,117)(96,116,113,99)(97,98,114,115) );
G=PermutationGroup([(1,93,124,110),(2,111,125,94),(3,95,126,112),(4,113,127,96),(5,97,128,114),(6,115,129,98),(7,99,130,116),(8,117,131,100),(9,81,132,118),(10,119,133,82),(11,83,134,120),(12,101,135,84),(13,85,136,102),(14,103,137,86),(15,87,138,104),(16,105,139,88),(17,89,140,106),(18,107,121,90),(19,91,122,108),(20,109,123,92),(21,68,153,41),(22,42,154,69),(23,70,155,43),(24,44,156,71),(25,72,157,45),(26,46,158,73),(27,74,159,47),(28,48,160,75),(29,76,141,49),(30,50,142,77),(31,78,143,51),(32,52,144,79),(33,80,145,53),(34,54,146,61),(35,62,147,55),(36,56,148,63),(37,64,149,57),(38,58,150,65),(39,66,151,59),(40,60,152,67)], [(1,36,134,158),(2,37,135,159),(3,38,136,160),(4,39,137,141),(5,40,138,142),(6,21,139,143),(7,22,140,144),(8,23,121,145),(9,24,122,146),(10,25,123,147),(11,26,124,148),(12,27,125,149),(13,28,126,150),(14,29,127,151),(15,30,128,152),(16,31,129,153),(17,32,130,154),(18,33,131,155),(19,34,132,156),(20,35,133,157),(41,105,78,98),(42,106,79,99),(43,107,80,100),(44,108,61,81),(45,109,62,82),(46,110,63,83),(47,111,64,84),(48,112,65,85),(49,113,66,86),(50,114,67,87),(51,115,68,88),(52,116,69,89),(53,117,70,90),(54,118,71,91),(55,119,72,92),(56,120,73,93),(57,101,74,94),(58,102,75,95),(59,103,76,96),(60,104,77,97)], [(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)], [(1,10,124,133),(2,132,125,9),(3,8,126,131),(4,130,127,7),(5,6,128,129),(11,20,134,123),(12,122,135,19),(13,18,136,121),(14,140,137,17),(15,16,138,139),(21,40,153,152),(22,151,154,39),(23,38,155,150),(24,149,156,37),(25,36,157,148),(26,147,158,35),(27,34,159,146),(28,145,160,33),(29,32,141,144),(30,143,142,31),(41,60,68,67),(42,66,69,59),(43,58,70,65),(44,64,71,57),(45,56,72,63),(46,62,73,55),(47,54,74,61),(48,80,75,53),(49,52,76,79),(50,78,77,51),(81,94,118,111),(82,110,119,93),(83,92,120,109),(84,108,101,91),(85,90,102,107),(86,106,103,89),(87,88,104,105),(95,100,112,117),(96,116,113,99),(97,98,114,115)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 14 | 11 |
| 0 | 0 | 0 | 31 | 14 | 14 |
| 0 | 0 | 21 | 4 | 10 | 0 |
| 0 | 0 | 20 | 21 | 0 | 10 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 15 | 37 | 0 | 0 | 0 | 0 |
| 36 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 32 | 11 |
| 0 | 0 | 34 | 40 | 11 | 32 |
| 0 | 0 | 19 | 28 | 1 | 34 |
| 0 | 0 | 23 | 19 | 7 | 34 |
| 26 | 4 | 0 | 0 | 0 | 0 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 32 | 11 |
| 0 | 0 | 40 | 34 | 11 | 14 |
| 0 | 0 | 28 | 19 | 1 | 0 |
| 0 | 0 | 19 | 23 | 7 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,0,21,20,0,0,0,31,4,21,0,0,14,14,10,0,0,0,11,14,0,10],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[15,36,0,0,0,0,37,26,0,0,0,0,0,0,7,34,19,23,0,0,7,40,28,19,0,0,32,11,1,7,0,0,11,32,34,34],[26,26,0,0,0,0,4,15,0,0,0,0,0,0,7,40,28,19,0,0,7,34,19,23,0,0,32,11,1,7,0,0,11,14,0,40] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | Q8.10D10 | D4.10D10 |
| kernel | C42.134D10 | C4×Dic10 | C20.6Q8 | C42⋊D5 | C42⋊2D5 | Dic5.Q8 | C4.Dic10 | D10⋊2Q8 | C4⋊C4⋊D5 | Dic5⋊Q8 | D10⋊3Q8 | Q8×C20 | C4×Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 6 | 6 | 2 | 16 | 2 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{134}D_{10} % in TeX
G:=Group("C4^2.134D10"); // GroupNames label
G:=SmallGroup(320,1255);
// by ID
G=gap.SmallGroup(320,1255);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^2,d^2=a^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,d*c*d^-1=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀