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), (C2×Dic5).210C23, (C4×Dic5).334C22, 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
Generators and relations for C42.102D10
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 >
Subgroups: 718 in 234 conjugacy classes, 101 normal (91 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C23.36C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, 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, C23.18D10, Dic5⋊D4, D4×C20, C42.102D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23.36C23, C4○D20, D4⋊2D5, C23×D5, C2×C4○D20, C2×D4⋊2D5, D5×C4○D4, C42.102D10
►On 160 points
Generators in S160
(1 65 60 106)(2 107 51 66)(3 67 52 108)(4 109 53 68)(5 69 54 110)(6 101 55 70)(7 61 56 102)(8 103 57 62)(9 63 58 104)(10 105 59 64)(11 146 151 41)(12 42 152 147)(13 148 153 43)(14 44 154 149)(15 150 155 45)(16 46 156 141)(17 142 157 47)(18 48 158 143)(19 144 159 49)(20 50 160 145)(21 36 126 131)(22 132 127 37)(23 38 128 133)(24 134 129 39)(25 40 130 135)(26 136 121 31)(27 32 122 137)(28 138 123 33)(29 34 124 139)(30 140 125 35)(71 82 112 99)(72 100 113 83)(73 84 114 91)(74 92 115 85)(75 86 116 93)(76 94 117 87)(77 88 118 95)(78 96 119 89)(79 90 120 97)(80 98 111 81)
(1 48 28 111)(2 49 29 112)(3 50 30 113)(4 41 21 114)(5 42 22 115)(6 43 23 116)(7 44 24 117)(8 45 25 118)(9 46 26 119)(10 47 27 120)(11 36 91 109)(12 37 92 110)(13 38 93 101)(14 39 94 102)(15 40 95 103)(16 31 96 104)(17 32 97 105)(18 33 98 106)(19 34 99 107)(20 35 100 108)(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)
(1 110 123 132)(2 109 124 131)(3 108 125 140)(4 107 126 139)(5 106 127 138)(6 105 128 137)(7 104 129 136)(8 103 130 135)(9 102 121 134)(10 101 122 133)(11 71 84 49)(12 80 85 48)(13 79 86 47)(14 78 87 46)(15 77 88 45)(16 76 89 44)(17 75 90 43)(18 74 81 42)(19 73 82 41)(20 72 83 50)(21 34 53 66)(22 33 54 65)(23 32 55 64)(24 31 56 63)(25 40 57 62)(26 39 58 61)(27 38 59 70)(28 37 60 69)(29 36 51 68)(30 35 52 67)(91 144 151 112)(92 143 152 111)(93 142 153 120)(94 141 154 119)(95 150 155 118)(96 149 156 117)(97 148 157 116)(98 147 158 115)(99 146 159 114)(100 145 160 113)
G:=sub<Sym(160)| (1,65,60,106)(2,107,51,66)(3,67,52,108)(4,109,53,68)(5,69,54,110)(6,101,55,70)(7,61,56,102)(8,103,57,62)(9,63,58,104)(10,105,59,64)(11,146,151,41)(12,42,152,147)(13,148,153,43)(14,44,154,149)(15,150,155,45)(16,46,156,141)(17,142,157,47)(18,48,158,143)(19,144,159,49)(20,50,160,145)(21,36,126,131)(22,132,127,37)(23,38,128,133)(24,134,129,39)(25,40,130,135)(26,136,121,31)(27,32,122,137)(28,138,123,33)(29,34,124,139)(30,140,125,35)(71,82,112,99)(72,100,113,83)(73,84,114,91)(74,92,115,85)(75,86,116,93)(76,94,117,87)(77,88,118,95)(78,96,119,89)(79,90,120,97)(80,98,111,81), (1,48,28,111)(2,49,29,112)(3,50,30,113)(4,41,21,114)(5,42,22,115)(6,43,23,116)(7,44,24,117)(8,45,25,118)(9,46,26,119)(10,47,27,120)(11,36,91,109)(12,37,92,110)(13,38,93,101)(14,39,94,102)(15,40,95,103)(16,31,96,104)(17,32,97,105)(18,33,98,106)(19,34,99,107)(20,35,100,108)(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), (1,110,123,132)(2,109,124,131)(3,108,125,140)(4,107,126,139)(5,106,127,138)(6,105,128,137)(7,104,129,136)(8,103,130,135)(9,102,121,134)(10,101,122,133)(11,71,84,49)(12,80,85,48)(13,79,86,47)(14,78,87,46)(15,77,88,45)(16,76,89,44)(17,75,90,43)(18,74,81,42)(19,73,82,41)(20,72,83,50)(21,34,53,66)(22,33,54,65)(23,32,55,64)(24,31,56,63)(25,40,57,62)(26,39,58,61)(27,38,59,70)(28,37,60,69)(29,36,51,68)(30,35,52,67)(91,144,151,112)(92,143,152,111)(93,142,153,120)(94,141,154,119)(95,150,155,118)(96,149,156,117)(97,148,157,116)(98,147,158,115)(99,146,159,114)(100,145,160,113)>;
G:=Group( (1,65,60,106)(2,107,51,66)(3,67,52,108)(4,109,53,68)(5,69,54,110)(6,101,55,70)(7,61,56,102)(8,103,57,62)(9,63,58,104)(10,105,59,64)(11,146,151,41)(12,42,152,147)(13,148,153,43)(14,44,154,149)(15,150,155,45)(16,46,156,141)(17,142,157,47)(18,48,158,143)(19,144,159,49)(20,50,160,145)(21,36,126,131)(22,132,127,37)(23,38,128,133)(24,134,129,39)(25,40,130,135)(26,136,121,31)(27,32,122,137)(28,138,123,33)(29,34,124,139)(30,140,125,35)(71,82,112,99)(72,100,113,83)(73,84,114,91)(74,92,115,85)(75,86,116,93)(76,94,117,87)(77,88,118,95)(78,96,119,89)(79,90,120,97)(80,98,111,81), (1,48,28,111)(2,49,29,112)(3,50,30,113)(4,41,21,114)(5,42,22,115)(6,43,23,116)(7,44,24,117)(8,45,25,118)(9,46,26,119)(10,47,27,120)(11,36,91,109)(12,37,92,110)(13,38,93,101)(14,39,94,102)(15,40,95,103)(16,31,96,104)(17,32,97,105)(18,33,98,106)(19,34,99,107)(20,35,100,108)(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), (1,110,123,132)(2,109,124,131)(3,108,125,140)(4,107,126,139)(5,106,127,138)(6,105,128,137)(7,104,129,136)(8,103,130,135)(9,102,121,134)(10,101,122,133)(11,71,84,49)(12,80,85,48)(13,79,86,47)(14,78,87,46)(15,77,88,45)(16,76,89,44)(17,75,90,43)(18,74,81,42)(19,73,82,41)(20,72,83,50)(21,34,53,66)(22,33,54,65)(23,32,55,64)(24,31,56,63)(25,40,57,62)(26,39,58,61)(27,38,59,70)(28,37,60,69)(29,36,51,68)(30,35,52,67)(91,144,151,112)(92,143,152,111)(93,142,153,120)(94,141,154,119)(95,150,155,118)(96,149,156,117)(97,148,157,116)(98,147,158,115)(99,146,159,114)(100,145,160,113) );
G=PermutationGroup([[(1,65,60,106),(2,107,51,66),(3,67,52,108),(4,109,53,68),(5,69,54,110),(6,101,55,70),(7,61,56,102),(8,103,57,62),(9,63,58,104),(10,105,59,64),(11,146,151,41),(12,42,152,147),(13,148,153,43),(14,44,154,149),(15,150,155,45),(16,46,156,141),(17,142,157,47),(18,48,158,143),(19,144,159,49),(20,50,160,145),(21,36,126,131),(22,132,127,37),(23,38,128,133),(24,134,129,39),(25,40,130,135),(26,136,121,31),(27,32,122,137),(28,138,123,33),(29,34,124,139),(30,140,125,35),(71,82,112,99),(72,100,113,83),(73,84,114,91),(74,92,115,85),(75,86,116,93),(76,94,117,87),(77,88,118,95),(78,96,119,89),(79,90,120,97),(80,98,111,81)], [(1,48,28,111),(2,49,29,112),(3,50,30,113),(4,41,21,114),(5,42,22,115),(6,43,23,116),(7,44,24,117),(8,45,25,118),(9,46,26,119),(10,47,27,120),(11,36,91,109),(12,37,92,110),(13,38,93,101),(14,39,94,102),(15,40,95,103),(16,31,96,104),(17,32,97,105),(18,33,98,106),(19,34,99,107),(20,35,100,108),(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)], [(1,110,123,132),(2,109,124,131),(3,108,125,140),(4,107,126,139),(5,106,127,138),(6,105,128,137),(7,104,129,136),(8,103,130,135),(9,102,121,134),(10,101,122,133),(11,71,84,49),(12,80,85,48),(13,79,86,47),(14,78,87,46),(15,77,88,45),(16,76,89,44),(17,75,90,43),(18,74,81,42),(19,73,82,41),(20,72,83,50),(21,34,53,66),(22,33,54,65),(23,32,55,64),(24,31,56,63),(25,40,57,62),(26,39,58,61),(27,38,59,70),(28,37,60,69),(29,36,51,68),(30,35,52,67),(91,144,151,112),(92,143,152,111),(93,142,153,120),(94,141,154,119),(95,150,155,118),(96,149,156,117),(97,148,157,116),(98,147,158,115),(99,146,159,114),(100,145,160,113)]])
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 |
Matrix representation of C42.102D10 ►in 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] >;
C42.102D10 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