metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.197D10, (D4×Dic5)⋊27C2, (C2×D4).159D10, (C2×C20).67C23, C22⋊C4.66D10, Dic5⋊4D4⋊17C2, Dic5⋊3Q8⋊30C2, (C2×C10).193C24, Dic5⋊D4.1C2, (C22×C4).321D10, C22.D4⋊17D5, C23.24(C22×D5), Dic5.14(C4○D4), Dic5.Q8⋊24C2, Dic5.5D4⋊30C2, (D4×C10).131C22, C23.D10⋊27C2, C4⋊Dic5.224C22, (C22×C10).29C23, (C2×Dic5).98C23, (C22×D5).84C23, C22.214(C23×D5), Dic5.14D4⋊28C2, C23.D5.39C22, D10⋊C4.31C22, C23.11D10⋊11C2, C23.23D10⋊19C2, C22.11(D4⋊2D5), (C22×C20).367C22, C5⋊8(C23.36C23), (C4×Dic5).128C22, C10.D4.38C22, (C2×Dic10).172C22, (C22×Dic5).249C22, (C2×C4×Dic5)⋊36C2, C2.57(D5×C4○D4), C4⋊C4⋊D5⋊26C2, C4⋊C4⋊7D5⋊31C2, C10.169(C2×C4○D4), C2.51(C2×D4⋊2D5), (C2×C4×D5).118C22, (C2×C4).58(C22×D5), (C2×C10).45(C4○D4), (C5×C4⋊C4).173C22, (C5×C22.D4)⋊3C2, (C2×C5⋊D4).45C22, (C5×C22⋊C4).48C22, SmallGroup(320,1321)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 734 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×14], C22, C22 [×2], C22 [×8], C5, C2×C4 [×5], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, D5, C10 [×3], C10 [×3], C42 [×6], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×4], Dic5 [×5], C20 [×5], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C42⋊2C2 [×2], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×7], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×C10 [×2], C23.36C23, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×3], C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C22×Dic5 [×3], C2×C5⋊D4 [×2], C22×C20, D4×C10, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊4D4 [×2], Dic5.5D4, Dic5⋊3Q8, Dic5.Q8, C4⋊C4⋊7D5, C4⋊C4⋊D5, C2×C4×Dic5, C23.23D10, D4×Dic5, Dic5⋊D4, C5×C22.D4, C4⋊C4.197D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D5×C4○D4 [×2], C4⋊C4.197D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2, bab-1=a-1, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 19 123 81)(2 50 124 159)(3 11 125 83)(4 42 126 151)(5 13 127 85)(6 44 128 153)(7 15 129 87)(8 46 130 155)(9 17 121 89)(10 48 122 157)(12 53 84 100)(14 55 86 92)(16 57 88 94)(18 59 90 96)(20 51 82 98)(21 36 140 145)(22 120 131 73)(23 38 132 147)(24 112 133 75)(25 40 134 149)(26 114 135 77)(27 32 136 141)(28 116 137 79)(29 34 138 143)(30 118 139 71)(31 62 150 109)(33 64 142 101)(35 66 144 103)(37 68 146 105)(39 70 148 107)(41 52 160 99)(43 54 152 91)(45 56 154 93)(47 58 156 95)(49 60 158 97)(61 76 108 113)(63 78 110 115)(65 80 102 117)(67 72 104 119)(69 74 106 111)
(1 117 97 34)(2 118 98 35)(3 119 99 36)(4 120 100 37)(5 111 91 38)(6 112 92 39)(7 113 93 40)(8 114 94 31)(9 115 95 32)(10 116 96 33)(11 104 41 21)(12 105 42 22)(13 106 43 23)(14 107 44 24)(15 108 45 25)(16 109 46 26)(17 110 47 27)(18 101 48 28)(19 102 49 29)(20 103 50 30)(51 144 124 71)(52 145 125 72)(53 146 126 73)(54 147 127 74)(55 148 128 75)(56 149 129 76)(57 150 130 77)(58 141 121 78)(59 142 122 79)(60 143 123 80)(61 154 134 87)(62 155 135 88)(63 156 136 89)(64 157 137 90)(65 158 138 81)(66 159 139 82)(67 160 140 83)(68 151 131 84)(69 152 132 85)(70 153 133 86)
(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 48 123 157)(2 47 124 156)(3 46 125 155)(4 45 126 154)(5 44 127 153)(6 43 128 152)(7 42 129 151)(8 41 130 160)(9 50 121 159)(10 49 122 158)(11 57 83 94)(12 56 84 93)(13 55 85 92)(14 54 86 91)(15 53 87 100)(16 52 88 99)(17 51 89 98)(18 60 90 97)(19 59 81 96)(20 58 82 95)(21 77 140 114)(22 76 131 113)(23 75 132 112)(24 74 133 111)(25 73 134 120)(26 72 135 119)(27 71 136 118)(28 80 137 117)(29 79 138 116)(30 78 139 115)(31 104 150 67)(32 103 141 66)(33 102 142 65)(34 101 143 64)(35 110 144 63)(36 109 145 62)(37 108 146 61)(38 107 147 70)(39 106 148 69)(40 105 149 68)
G:=sub<Sym(160)| (1,19,123,81)(2,50,124,159)(3,11,125,83)(4,42,126,151)(5,13,127,85)(6,44,128,153)(7,15,129,87)(8,46,130,155)(9,17,121,89)(10,48,122,157)(12,53,84,100)(14,55,86,92)(16,57,88,94)(18,59,90,96)(20,51,82,98)(21,36,140,145)(22,120,131,73)(23,38,132,147)(24,112,133,75)(25,40,134,149)(26,114,135,77)(27,32,136,141)(28,116,137,79)(29,34,138,143)(30,118,139,71)(31,62,150,109)(33,64,142,101)(35,66,144,103)(37,68,146,105)(39,70,148,107)(41,52,160,99)(43,54,152,91)(45,56,154,93)(47,58,156,95)(49,60,158,97)(61,76,108,113)(63,78,110,115)(65,80,102,117)(67,72,104,119)(69,74,106,111), (1,117,97,34)(2,118,98,35)(3,119,99,36)(4,120,100,37)(5,111,91,38)(6,112,92,39)(7,113,93,40)(8,114,94,31)(9,115,95,32)(10,116,96,33)(11,104,41,21)(12,105,42,22)(13,106,43,23)(14,107,44,24)(15,108,45,25)(16,109,46,26)(17,110,47,27)(18,101,48,28)(19,102,49,29)(20,103,50,30)(51,144,124,71)(52,145,125,72)(53,146,126,73)(54,147,127,74)(55,148,128,75)(56,149,129,76)(57,150,130,77)(58,141,121,78)(59,142,122,79)(60,143,123,80)(61,154,134,87)(62,155,135,88)(63,156,136,89)(64,157,137,90)(65,158,138,81)(66,159,139,82)(67,160,140,83)(68,151,131,84)(69,152,132,85)(70,153,133,86), (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), (1,48,123,157)(2,47,124,156)(3,46,125,155)(4,45,126,154)(5,44,127,153)(6,43,128,152)(7,42,129,151)(8,41,130,160)(9,50,121,159)(10,49,122,158)(11,57,83,94)(12,56,84,93)(13,55,85,92)(14,54,86,91)(15,53,87,100)(16,52,88,99)(17,51,89,98)(18,60,90,97)(19,59,81,96)(20,58,82,95)(21,77,140,114)(22,76,131,113)(23,75,132,112)(24,74,133,111)(25,73,134,120)(26,72,135,119)(27,71,136,118)(28,80,137,117)(29,79,138,116)(30,78,139,115)(31,104,150,67)(32,103,141,66)(33,102,142,65)(34,101,143,64)(35,110,144,63)(36,109,145,62)(37,108,146,61)(38,107,147,70)(39,106,148,69)(40,105,149,68)>;
G:=Group( (1,19,123,81)(2,50,124,159)(3,11,125,83)(4,42,126,151)(5,13,127,85)(6,44,128,153)(7,15,129,87)(8,46,130,155)(9,17,121,89)(10,48,122,157)(12,53,84,100)(14,55,86,92)(16,57,88,94)(18,59,90,96)(20,51,82,98)(21,36,140,145)(22,120,131,73)(23,38,132,147)(24,112,133,75)(25,40,134,149)(26,114,135,77)(27,32,136,141)(28,116,137,79)(29,34,138,143)(30,118,139,71)(31,62,150,109)(33,64,142,101)(35,66,144,103)(37,68,146,105)(39,70,148,107)(41,52,160,99)(43,54,152,91)(45,56,154,93)(47,58,156,95)(49,60,158,97)(61,76,108,113)(63,78,110,115)(65,80,102,117)(67,72,104,119)(69,74,106,111), (1,117,97,34)(2,118,98,35)(3,119,99,36)(4,120,100,37)(5,111,91,38)(6,112,92,39)(7,113,93,40)(8,114,94,31)(9,115,95,32)(10,116,96,33)(11,104,41,21)(12,105,42,22)(13,106,43,23)(14,107,44,24)(15,108,45,25)(16,109,46,26)(17,110,47,27)(18,101,48,28)(19,102,49,29)(20,103,50,30)(51,144,124,71)(52,145,125,72)(53,146,126,73)(54,147,127,74)(55,148,128,75)(56,149,129,76)(57,150,130,77)(58,141,121,78)(59,142,122,79)(60,143,123,80)(61,154,134,87)(62,155,135,88)(63,156,136,89)(64,157,137,90)(65,158,138,81)(66,159,139,82)(67,160,140,83)(68,151,131,84)(69,152,132,85)(70,153,133,86), (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), (1,48,123,157)(2,47,124,156)(3,46,125,155)(4,45,126,154)(5,44,127,153)(6,43,128,152)(7,42,129,151)(8,41,130,160)(9,50,121,159)(10,49,122,158)(11,57,83,94)(12,56,84,93)(13,55,85,92)(14,54,86,91)(15,53,87,100)(16,52,88,99)(17,51,89,98)(18,60,90,97)(19,59,81,96)(20,58,82,95)(21,77,140,114)(22,76,131,113)(23,75,132,112)(24,74,133,111)(25,73,134,120)(26,72,135,119)(27,71,136,118)(28,80,137,117)(29,79,138,116)(30,78,139,115)(31,104,150,67)(32,103,141,66)(33,102,142,65)(34,101,143,64)(35,110,144,63)(36,109,145,62)(37,108,146,61)(38,107,147,70)(39,106,148,69)(40,105,149,68) );
G=PermutationGroup([(1,19,123,81),(2,50,124,159),(3,11,125,83),(4,42,126,151),(5,13,127,85),(6,44,128,153),(7,15,129,87),(8,46,130,155),(9,17,121,89),(10,48,122,157),(12,53,84,100),(14,55,86,92),(16,57,88,94),(18,59,90,96),(20,51,82,98),(21,36,140,145),(22,120,131,73),(23,38,132,147),(24,112,133,75),(25,40,134,149),(26,114,135,77),(27,32,136,141),(28,116,137,79),(29,34,138,143),(30,118,139,71),(31,62,150,109),(33,64,142,101),(35,66,144,103),(37,68,146,105),(39,70,148,107),(41,52,160,99),(43,54,152,91),(45,56,154,93),(47,58,156,95),(49,60,158,97),(61,76,108,113),(63,78,110,115),(65,80,102,117),(67,72,104,119),(69,74,106,111)], [(1,117,97,34),(2,118,98,35),(3,119,99,36),(4,120,100,37),(5,111,91,38),(6,112,92,39),(7,113,93,40),(8,114,94,31),(9,115,95,32),(10,116,96,33),(11,104,41,21),(12,105,42,22),(13,106,43,23),(14,107,44,24),(15,108,45,25),(16,109,46,26),(17,110,47,27),(18,101,48,28),(19,102,49,29),(20,103,50,30),(51,144,124,71),(52,145,125,72),(53,146,126,73),(54,147,127,74),(55,148,128,75),(56,149,129,76),(57,150,130,77),(58,141,121,78),(59,142,122,79),(60,143,123,80),(61,154,134,87),(62,155,135,88),(63,156,136,89),(64,157,137,90),(65,158,138,81),(66,159,139,82),(67,160,140,83),(68,151,131,84),(69,152,132,85),(70,153,133,86)], [(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)], [(1,48,123,157),(2,47,124,156),(3,46,125,155),(4,45,126,154),(5,44,127,153),(6,43,128,152),(7,42,129,151),(8,41,130,160),(9,50,121,159),(10,49,122,158),(11,57,83,94),(12,56,84,93),(13,55,85,92),(14,54,86,91),(15,53,87,100),(16,52,88,99),(17,51,89,98),(18,60,90,97),(19,59,81,96),(20,58,82,95),(21,77,140,114),(22,76,131,113),(23,75,132,112),(24,74,133,111),(25,73,134,120),(26,72,135,119),(27,71,136,118),(28,80,137,117),(29,79,138,116),(30,78,139,115),(31,104,150,67),(32,103,141,66),(33,102,142,65),(34,101,143,64),(35,110,144,63),(36,109,145,62),(37,108,146,61),(38,107,147,70),(39,106,148,69),(40,105,149,68)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 23 | 40 |
| 9 | 36 | 0 | 0 | 0 | 0 |
| 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 0 | 0 | 2 | 32 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 40 |
| 0 | 0 | 0 | 0 | 39 | 9 |
G:=sub<GL(6,GF(41))| [9,16,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,23,0,0,0,0,0,40],[9,16,0,0,0,0,36,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,9,2,0,0,0,0,1,32],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,32,39,0,0,0,0,40,9] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D4⋊2D5 | D5×C4○D4 |
| kernel | C4⋊C4.197D10 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | Dic5.5D4 | Dic5⋊3Q8 | Dic5.Q8 | C4⋊C4⋊7D5 | C4⋊C4⋊D5 | C2×C4×Dic5 | C23.23D10 | D4×Dic5 | Dic5⋊D4 | C5×C22.D4 | C22.D4 | Dic5 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 4 | 6 | 4 | 2 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._{197}D_{10} % in TeX
G:=Group("C4:C4.197D10"); // GroupNames label
G:=SmallGroup(320,1321);
// by ID
G=gap.SmallGroup(320,1321);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,346,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀