metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.102D10, (C4×D4)⋊5D5, (D4×C20)⋊6C2, C4⋊C4.277D10, (C4×Dic10)⋊25C2, (C2×D4).207D10, C42⋊D5⋊10C2, (C2×C10).82C24, C20.309(C4○D4), (C4×C20).145C22, (C2×C20).584C23, C22⋊C4.105D10, Dic5.9(C4○D4), Dic5⋊D4.6C2, C22.5(C4○D20), (C22×C4).320D10, C4.136(D4⋊2D5), C23.91(C22×D5), Dic5.Q8⋊45C2, Dic5.5D4⋊49C2, (D4×C10).301C22, C22.D20⋊33C2, C23.21D10⋊5C2, C23.D10⋊49C2, C4⋊Dic5.295C22, (C22×D5).28C23, C22.110(C23×D5), Dic5.14D4⋊49C2, D10⋊C4.97C22, C23.18D10⋊33C2, (C22×C10).152C23, (C22×C20).104C22, C5⋊3(C23.36C23), (C4×Dic5).334C22, (C2×Dic5).210C23, C23.D5.101C22, (C2×Dic10).243C22, C10.D4.107C22, (C22×Dic5).242C22, (C4×C5⋊D4)⋊3C2, (C2×C4×Dic5)⋊35C2, C2.17(D5×C4○D4), C4⋊C4⋊D5⋊50C2, C2.39(C2×C4○D20), C10.136(C2×C4○D4), C2.18(C2×D4⋊2D5), (C2×C4×D5).248C22, (C2×C10).13(C4○D4), (C5×C4⋊C4).318C22, (C2×C4).153(C22×D5), (C2×C5⋊D4).116C22, (C5×C22⋊C4).119C22, SmallGroup(320,1210)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 718 in 234 conjugacy classes, 101 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], 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, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×4], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×7], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5, C22×C10 [×2], C23.36C23, C4×Dic5 [×5], C10.D4 [×6], C4⋊Dic5 [×3], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C42⋊D5, Dic5.14D4, C23.D10, Dic5.5D4, C22.D20, Dic5.Q8, C4⋊C4⋊D5, C2×C4×Dic5, C23.21D10, C4×C5⋊D4 [×2], C23.18D10, Dic5⋊D4, D4×C20, C42.102D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C4○D20 [×2], D4⋊2D5 [×2], C23×D5, C2×C4○D20, C2×D4⋊2D5, D5×C4○D4, C42.102D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 65 60 29)(2 30 51 66)(3 67 52 21)(4 22 53 68)(5 69 54 23)(6 24 55 70)(7 61 56 25)(8 26 57 62)(9 63 58 27)(10 28 59 64)(11 144 159 118)(12 119 160 145)(13 146 151 120)(14 111 152 147)(15 148 153 112)(16 113 154 149)(17 150 155 114)(18 115 156 141)(19 142 157 116)(20 117 158 143)(31 46 77 88)(32 89 78 47)(33 48 79 90)(34 81 80 49)(35 50 71 82)(36 83 72 41)(37 42 73 84)(38 85 74 43)(39 44 75 86)(40 87 76 45)(91 132 127 106)(92 107 128 133)(93 134 129 108)(94 109 130 135)(95 136 121 110)(96 101 122 137)(97 138 123 102)(98 103 124 139)(99 140 125 104)(100 105 126 131)
(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)(11 103 50 30)(12 104 41 21)(13 105 42 22)(14 106 43 23)(15 107 44 24)(16 108 45 25)(17 109 46 26)(18 110 47 27)(19 101 48 28)(20 102 49 29)(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 23 123 132)(2 22 124 131)(3 21 125 140)(4 30 126 139)(5 29 127 138)(6 28 128 137)(7 27 129 136)(8 26 130 135)(9 25 121 134)(10 24 122 133)(11 73 82 120)(12 72 83 119)(13 71 84 118)(14 80 85 117)(15 79 86 116)(16 78 87 115)(17 77 88 114)(18 76 89 113)(19 75 90 112)(20 74 81 111)(31 46 150 155)(32 45 141 154)(33 44 142 153)(34 43 143 152)(35 42 144 151)(36 41 145 160)(37 50 146 159)(38 49 147 158)(39 48 148 157)(40 47 149 156)(51 68 98 105)(52 67 99 104)(53 66 100 103)(54 65 91 102)(55 64 92 101)(56 63 93 110)(57 62 94 109)(58 61 95 108)(59 70 96 107)(60 69 97 106)
G:=sub<Sym(160)| (1,65,60,29)(2,30,51,66)(3,67,52,21)(4,22,53,68)(5,69,54,23)(6,24,55,70)(7,61,56,25)(8,26,57,62)(9,63,58,27)(10,28,59,64)(11,144,159,118)(12,119,160,145)(13,146,151,120)(14,111,152,147)(15,148,153,112)(16,113,154,149)(17,150,155,114)(18,115,156,141)(19,142,157,116)(20,117,158,143)(31,46,77,88)(32,89,78,47)(33,48,79,90)(34,81,80,49)(35,50,71,82)(36,83,72,41)(37,42,73,84)(38,85,74,43)(39,44,75,86)(40,87,76,45)(91,132,127,106)(92,107,128,133)(93,134,129,108)(94,109,130,135)(95,136,121,110)(96,101,122,137)(97,138,123,102)(98,103,124,139)(99,140,125,104)(100,105,126,131), (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)(11,103,50,30)(12,104,41,21)(13,105,42,22)(14,106,43,23)(15,107,44,24)(16,108,45,25)(17,109,46,26)(18,110,47,27)(19,101,48,28)(20,102,49,29)(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), (1,23,123,132)(2,22,124,131)(3,21,125,140)(4,30,126,139)(5,29,127,138)(6,28,128,137)(7,27,129,136)(8,26,130,135)(9,25,121,134)(10,24,122,133)(11,73,82,120)(12,72,83,119)(13,71,84,118)(14,80,85,117)(15,79,86,116)(16,78,87,115)(17,77,88,114)(18,76,89,113)(19,75,90,112)(20,74,81,111)(31,46,150,155)(32,45,141,154)(33,44,142,153)(34,43,143,152)(35,42,144,151)(36,41,145,160)(37,50,146,159)(38,49,147,158)(39,48,148,157)(40,47,149,156)(51,68,98,105)(52,67,99,104)(53,66,100,103)(54,65,91,102)(55,64,92,101)(56,63,93,110)(57,62,94,109)(58,61,95,108)(59,70,96,107)(60,69,97,106)>;
G:=Group( (1,65,60,29)(2,30,51,66)(3,67,52,21)(4,22,53,68)(5,69,54,23)(6,24,55,70)(7,61,56,25)(8,26,57,62)(9,63,58,27)(10,28,59,64)(11,144,159,118)(12,119,160,145)(13,146,151,120)(14,111,152,147)(15,148,153,112)(16,113,154,149)(17,150,155,114)(18,115,156,141)(19,142,157,116)(20,117,158,143)(31,46,77,88)(32,89,78,47)(33,48,79,90)(34,81,80,49)(35,50,71,82)(36,83,72,41)(37,42,73,84)(38,85,74,43)(39,44,75,86)(40,87,76,45)(91,132,127,106)(92,107,128,133)(93,134,129,108)(94,109,130,135)(95,136,121,110)(96,101,122,137)(97,138,123,102)(98,103,124,139)(99,140,125,104)(100,105,126,131), (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)(11,103,50,30)(12,104,41,21)(13,105,42,22)(14,106,43,23)(15,107,44,24)(16,108,45,25)(17,109,46,26)(18,110,47,27)(19,101,48,28)(20,102,49,29)(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), (1,23,123,132)(2,22,124,131)(3,21,125,140)(4,30,126,139)(5,29,127,138)(6,28,128,137)(7,27,129,136)(8,26,130,135)(9,25,121,134)(10,24,122,133)(11,73,82,120)(12,72,83,119)(13,71,84,118)(14,80,85,117)(15,79,86,116)(16,78,87,115)(17,77,88,114)(18,76,89,113)(19,75,90,112)(20,74,81,111)(31,46,150,155)(32,45,141,154)(33,44,142,153)(34,43,143,152)(35,42,144,151)(36,41,145,160)(37,50,146,159)(38,49,147,158)(39,48,148,157)(40,47,149,156)(51,68,98,105)(52,67,99,104)(53,66,100,103)(54,65,91,102)(55,64,92,101)(56,63,93,110)(57,62,94,109)(58,61,95,108)(59,70,96,107)(60,69,97,106) );
G=PermutationGroup([(1,65,60,29),(2,30,51,66),(3,67,52,21),(4,22,53,68),(5,69,54,23),(6,24,55,70),(7,61,56,25),(8,26,57,62),(9,63,58,27),(10,28,59,64),(11,144,159,118),(12,119,160,145),(13,146,151,120),(14,111,152,147),(15,148,153,112),(16,113,154,149),(17,150,155,114),(18,115,156,141),(19,142,157,116),(20,117,158,143),(31,46,77,88),(32,89,78,47),(33,48,79,90),(34,81,80,49),(35,50,71,82),(36,83,72,41),(37,42,73,84),(38,85,74,43),(39,44,75,86),(40,87,76,45),(91,132,127,106),(92,107,128,133),(93,134,129,108),(94,109,130,135),(95,136,121,110),(96,101,122,137),(97,138,123,102),(98,103,124,139),(99,140,125,104),(100,105,126,131)], [(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),(11,103,50,30),(12,104,41,21),(13,105,42,22),(14,106,43,23),(15,107,44,24),(16,108,45,25),(17,109,46,26),(18,110,47,27),(19,101,48,28),(20,102,49,29),(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)], [(1,23,123,132),(2,22,124,131),(3,21,125,140),(4,30,126,139),(5,29,127,138),(6,28,128,137),(7,27,129,136),(8,26,130,135),(9,25,121,134),(10,24,122,133),(11,73,82,120),(12,72,83,119),(13,71,84,118),(14,80,85,117),(15,79,86,116),(16,78,87,115),(17,77,88,114),(18,76,89,113),(19,75,90,112),(20,74,81,111),(31,46,150,155),(32,45,141,154),(33,44,142,153),(34,43,143,152),(35,42,144,151),(36,41,145,160),(37,50,146,159),(38,49,147,158),(39,48,148,157),(40,47,149,156),(51,68,98,105),(52,67,99,104),(53,66,100,103),(54,65,91,102),(55,64,92,101),(56,63,93,110),(57,62,94,109),(58,61,95,108),(59,70,96,107),(60,69,97,106)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 23 | 6 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 32 | 9 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 35 | 35 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 21 | 40 |
| 0 | 0 | 30 | 20 |
| 21 | 20 | 0 | 0 |
| 23 | 20 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [23,35,0,0,6,18,0,0,0,0,32,32,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[35,6,0,0,35,40,0,0,0,0,21,30,0,0,40,20],[21,23,0,0,20,20,0,0,0,0,9,0,0,0,0,9] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 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 | 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 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | D4⋊2D5 | D5×C4○D4 |
| kernel | C42.102D10 | C4×Dic10 | C42⋊D5 | Dic5.14D4 | C23.D10 | Dic5.5D4 | C22.D20 | Dic5.Q8 | C4⋊C4⋊D5 | C2×C4×Dic5 | C23.21D10 | C4×C5⋊D4 | C23.18D10 | Dic5⋊D4 | D4×C20 | C4×D4 | Dic5 | C20 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{102}D_{10} % in TeX
G:=Group("C4^2.102D10"); // GroupNames label
G:=SmallGroup(320,1210);
// by ID
G=gap.SmallGroup(320,1210);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1123,794,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-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
○
≀