metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.137D10, C10.702+ (1+4), C10.872- (1+4), C4.4D4⋊6D5, C42⋊2D5⋊7C2, (C2×Q8).81D10, D10⋊3Q8⋊27C2, (C4×Dic10)⋊43C2, (C2×D4).107D10, C22⋊C4.71D10, Dic5⋊4D4⋊28C2, Dic5⋊Q8⋊20C2, (C2×C10).213C24, (C4×C20).183C22, (C2×C20).629C23, Dic5⋊D4.5C2, D10.12D4⋊40C2, C2.72(D4⋊6D10), C23.35(C22×D5), Dic5.17(C4○D4), Dic5.5D4⋊37C2, (D4×C10).207C22, C23.D10⋊36C2, C4⋊Dic5.232C22, (C22×C10).43C23, (Q8×C10).122C22, (C22×D5).93C23, C22.234(C23×D5), Dic5.14D4⋊37C2, C23.D5.50C22, D10⋊C4.59C22, C23.18D10⋊24C2, C23.11D10⋊16C2, C5⋊8(C22.36C24), (C2×Dic5).260C23, (C4×Dic5).233C22, C10.D4.82C22, C2.48(D4.10D10), (C2×Dic10).180C22, (C22×Dic5).138C22, C2.72(D5×C4○D4), (C5×C4.4D4)⋊7C2, C10.184(C2×C4○D4), (C2×C4×D5).128C22, (C2×C4).191(C22×D5), (C2×C5⋊D4).56C22, (C5×C22⋊C4).60C22, SmallGroup(320,1341)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 734 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C5, C2×C4 [×5], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, D5, C10 [×3], C10 [×2], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×5], D10 [×3], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C4.4D4 [×2], C42⋊2C2 [×2], C4⋊Q8, Dic10 [×3], C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×5], C5×D4, C5×Q8, C22×D5, C22×C10 [×2], C22.36C24, C4×Dic5 [×3], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C4×Dic10, C42⋊2D5, C23.11D10, Dic5.14D4 [×2], C23.D10, Dic5⋊4D4, D10.12D4, Dic5.5D4 [×2], C23.18D10, Dic5⋊D4, Dic5⋊Q8, D10⋊3Q8, C5×C4.4D4, C42.137D10
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], C22.36C24, C23×D5, D4⋊6D10, D5×C4○D4, D4.10D10, C42.137D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 115 95 32)(2 71 96 144)(3 117 97 34)(4 73 98 146)(5 119 99 36)(6 75 100 148)(7 111 91 38)(8 77 92 150)(9 113 93 40)(10 79 94 142)(11 69 41 132)(12 105 42 22)(13 61 43 134)(14 107 44 24)(15 63 45 136)(16 109 46 26)(17 65 47 138)(18 101 48 28)(19 67 49 140)(20 103 50 30)(21 85 104 152)(23 87 106 154)(25 89 108 156)(27 81 110 158)(29 83 102 160)(31 122 114 59)(33 124 116 51)(35 126 118 53)(37 128 120 55)(39 130 112 57)(52 145 125 72)(54 147 127 74)(56 149 129 76)(58 141 121 78)(60 143 123 80)(62 155 135 88)(64 157 137 90)(66 159 139 82)(68 151 131 84)(70 153 133 86)
(1 133 123 22)(2 61 124 106)(3 135 125 24)(4 63 126 108)(5 137 127 26)(6 65 128 110)(7 139 129 28)(8 67 130 102)(9 131 121 30)(10 69 122 104)(11 31 85 142)(12 115 86 80)(13 33 87 144)(14 117 88 72)(15 35 89 146)(16 119 90 74)(17 37 81 148)(18 111 82 76)(19 39 83 150)(20 113 84 78)(21 94 132 59)(23 96 134 51)(25 98 136 53)(27 100 138 55)(29 92 140 57)(32 153 143 42)(34 155 145 44)(36 157 147 46)(38 159 149 48)(40 151 141 50)(41 114 152 79)(43 116 154 71)(45 118 156 73)(47 120 158 75)(49 112 160 77)(52 107 97 62)(54 109 99 64)(56 101 91 66)(58 103 93 68)(60 105 95 70)
(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 46 123 157)(2 45 124 156)(3 44 125 155)(4 43 126 154)(5 42 127 153)(6 41 128 152)(7 50 129 151)(8 49 130 160)(9 48 121 159)(10 47 122 158)(11 55 85 100)(12 54 86 99)(13 53 87 98)(14 52 88 97)(15 51 89 96)(16 60 90 95)(17 59 81 94)(18 58 82 93)(19 57 83 92)(20 56 84 91)(21 120 132 75)(22 119 133 74)(23 118 134 73)(24 117 135 72)(25 116 136 71)(26 115 137 80)(27 114 138 79)(28 113 139 78)(29 112 140 77)(30 111 131 76)(31 65 142 110)(32 64 143 109)(33 63 144 108)(34 62 145 107)(35 61 146 106)(36 70 147 105)(37 69 148 104)(38 68 149 103)(39 67 150 102)(40 66 141 101)
G:=sub<Sym(160)| (1,115,95,32)(2,71,96,144)(3,117,97,34)(4,73,98,146)(5,119,99,36)(6,75,100,148)(7,111,91,38)(8,77,92,150)(9,113,93,40)(10,79,94,142)(11,69,41,132)(12,105,42,22)(13,61,43,134)(14,107,44,24)(15,63,45,136)(16,109,46,26)(17,65,47,138)(18,101,48,28)(19,67,49,140)(20,103,50,30)(21,85,104,152)(23,87,106,154)(25,89,108,156)(27,81,110,158)(29,83,102,160)(31,122,114,59)(33,124,116,51)(35,126,118,53)(37,128,120,55)(39,130,112,57)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,22)(2,61,124,106)(3,135,125,24)(4,63,126,108)(5,137,127,26)(6,65,128,110)(7,139,129,28)(8,67,130,102)(9,131,121,30)(10,69,122,104)(11,31,85,142)(12,115,86,80)(13,33,87,144)(14,117,88,72)(15,35,89,146)(16,119,90,74)(17,37,81,148)(18,111,82,76)(19,39,83,150)(20,113,84,78)(21,94,132,59)(23,96,134,51)(25,98,136,53)(27,100,138,55)(29,92,140,57)(32,153,143,42)(34,155,145,44)(36,157,147,46)(38,159,149,48)(40,151,141,50)(41,114,152,79)(43,116,154,71)(45,118,156,73)(47,120,158,75)(49,112,160,77)(52,107,97,62)(54,109,99,64)(56,101,91,66)(58,103,93,68)(60,105,95,70), (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), (1,46,123,157)(2,45,124,156)(3,44,125,155)(4,43,126,154)(5,42,127,153)(6,41,128,152)(7,50,129,151)(8,49,130,160)(9,48,121,159)(10,47,122,158)(11,55,85,100)(12,54,86,99)(13,53,87,98)(14,52,88,97)(15,51,89,96)(16,60,90,95)(17,59,81,94)(18,58,82,93)(19,57,83,92)(20,56,84,91)(21,120,132,75)(22,119,133,74)(23,118,134,73)(24,117,135,72)(25,116,136,71)(26,115,137,80)(27,114,138,79)(28,113,139,78)(29,112,140,77)(30,111,131,76)(31,65,142,110)(32,64,143,109)(33,63,144,108)(34,62,145,107)(35,61,146,106)(36,70,147,105)(37,69,148,104)(38,68,149,103)(39,67,150,102)(40,66,141,101)>;
G:=Group( (1,115,95,32)(2,71,96,144)(3,117,97,34)(4,73,98,146)(5,119,99,36)(6,75,100,148)(7,111,91,38)(8,77,92,150)(9,113,93,40)(10,79,94,142)(11,69,41,132)(12,105,42,22)(13,61,43,134)(14,107,44,24)(15,63,45,136)(16,109,46,26)(17,65,47,138)(18,101,48,28)(19,67,49,140)(20,103,50,30)(21,85,104,152)(23,87,106,154)(25,89,108,156)(27,81,110,158)(29,83,102,160)(31,122,114,59)(33,124,116,51)(35,126,118,53)(37,128,120,55)(39,130,112,57)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,22)(2,61,124,106)(3,135,125,24)(4,63,126,108)(5,137,127,26)(6,65,128,110)(7,139,129,28)(8,67,130,102)(9,131,121,30)(10,69,122,104)(11,31,85,142)(12,115,86,80)(13,33,87,144)(14,117,88,72)(15,35,89,146)(16,119,90,74)(17,37,81,148)(18,111,82,76)(19,39,83,150)(20,113,84,78)(21,94,132,59)(23,96,134,51)(25,98,136,53)(27,100,138,55)(29,92,140,57)(32,153,143,42)(34,155,145,44)(36,157,147,46)(38,159,149,48)(40,151,141,50)(41,114,152,79)(43,116,154,71)(45,118,156,73)(47,120,158,75)(49,112,160,77)(52,107,97,62)(54,109,99,64)(56,101,91,66)(58,103,93,68)(60,105,95,70), (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), (1,46,123,157)(2,45,124,156)(3,44,125,155)(4,43,126,154)(5,42,127,153)(6,41,128,152)(7,50,129,151)(8,49,130,160)(9,48,121,159)(10,47,122,158)(11,55,85,100)(12,54,86,99)(13,53,87,98)(14,52,88,97)(15,51,89,96)(16,60,90,95)(17,59,81,94)(18,58,82,93)(19,57,83,92)(20,56,84,91)(21,120,132,75)(22,119,133,74)(23,118,134,73)(24,117,135,72)(25,116,136,71)(26,115,137,80)(27,114,138,79)(28,113,139,78)(29,112,140,77)(30,111,131,76)(31,65,142,110)(32,64,143,109)(33,63,144,108)(34,62,145,107)(35,61,146,106)(36,70,147,105)(37,69,148,104)(38,68,149,103)(39,67,150,102)(40,66,141,101) );
G=PermutationGroup([(1,115,95,32),(2,71,96,144),(3,117,97,34),(4,73,98,146),(5,119,99,36),(6,75,100,148),(7,111,91,38),(8,77,92,150),(9,113,93,40),(10,79,94,142),(11,69,41,132),(12,105,42,22),(13,61,43,134),(14,107,44,24),(15,63,45,136),(16,109,46,26),(17,65,47,138),(18,101,48,28),(19,67,49,140),(20,103,50,30),(21,85,104,152),(23,87,106,154),(25,89,108,156),(27,81,110,158),(29,83,102,160),(31,122,114,59),(33,124,116,51),(35,126,118,53),(37,128,120,55),(39,130,112,57),(52,145,125,72),(54,147,127,74),(56,149,129,76),(58,141,121,78),(60,143,123,80),(62,155,135,88),(64,157,137,90),(66,159,139,82),(68,151,131,84),(70,153,133,86)], [(1,133,123,22),(2,61,124,106),(3,135,125,24),(4,63,126,108),(5,137,127,26),(6,65,128,110),(7,139,129,28),(8,67,130,102),(9,131,121,30),(10,69,122,104),(11,31,85,142),(12,115,86,80),(13,33,87,144),(14,117,88,72),(15,35,89,146),(16,119,90,74),(17,37,81,148),(18,111,82,76),(19,39,83,150),(20,113,84,78),(21,94,132,59),(23,96,134,51),(25,98,136,53),(27,100,138,55),(29,92,140,57),(32,153,143,42),(34,155,145,44),(36,157,147,46),(38,159,149,48),(40,151,141,50),(41,114,152,79),(43,116,154,71),(45,118,156,73),(47,120,158,75),(49,112,160,77),(52,107,97,62),(54,109,99,64),(56,101,91,66),(58,103,93,68),(60,105,95,70)], [(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)], [(1,46,123,157),(2,45,124,156),(3,44,125,155),(4,43,126,154),(5,42,127,153),(6,41,128,152),(7,50,129,151),(8,49,130,160),(9,48,121,159),(10,47,122,158),(11,55,85,100),(12,54,86,99),(13,53,87,98),(14,52,88,97),(15,51,89,96),(16,60,90,95),(17,59,81,94),(18,58,82,93),(19,57,83,92),(20,56,84,91),(21,120,132,75),(22,119,133,74),(23,118,134,73),(24,117,135,72),(25,116,136,71),(26,115,137,80),(27,114,138,79),(28,113,139,78),(29,112,140,77),(30,111,131,76),(31,65,142,110),(32,64,143,109),(33,63,144,108),(34,62,145,107),(35,61,146,106),(36,70,147,105),(37,69,148,104),(38,68,149,103),(39,67,150,102),(40,66,141,101)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 10 | 28 |
| 0 | 0 | 6 | 23 | 10 | 23 |
| 0 | 0 | 37 | 12 | 24 | 6 |
| 0 | 0 | 16 | 25 | 34 | 17 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 38 | 38 |
| 0 | 0 | 0 | 1 | 3 | 0 |
| 0 | 0 | 0 | 13 | 40 | 0 |
| 0 | 0 | 28 | 28 | 0 | 40 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 30 |
| 0 | 0 | 20 | 24 | 6 | 11 |
| 0 | 0 | 7 | 33 | 37 | 20 |
| 0 | 0 | 0 | 15 | 4 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 21 | 0 | 0 |
| 0 | 0 | 18 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 23 |
| 0 | 0 | 0 | 0 | 37 | 38 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,18,6,37,16,0,0,35,23,12,25,0,0,10,10,24,34,0,0,28,23,6,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,28,0,0,0,1,13,28,0,0,38,3,40,0,0,0,38,0,0,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,21,20,7,0,0,0,21,24,33,15,0,0,0,6,37,4,0,0,30,11,20,0],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,20,18,0,0,0,0,21,21,0,0,0,0,0,0,3,37,0,0,0,0,23,38] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D5×C4○D4 | D4.10D10 |
| kernel | C42.137D10 | C4×Dic10 | C42⋊2D5 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C23.18D10 | Dic5⋊D4 | Dic5⋊Q8 | D10⋊3Q8 | C5×C4.4D4 | C4.4D4 | Dic5 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{137}D_{10} % in TeX
G:=Group("C4^2.137D10"); // GroupNames label
G:=SmallGroup(320,1341);
// by ID
G=gap.SmallGroup(320,1341);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,1123,346,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀