metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.150D10, C10.282- (1+4), C10.1322+ (1+4), (C4×D20)⋊47C2, C42.C2⋊6D5, C4⋊C4.207D10, C42⋊2D5⋊9C2, D20⋊8C4⋊36C2, D10⋊2Q8⋊37C2, D10⋊Q8⋊35C2, C4⋊2D20.13C2, D10.17(C4○D4), (C2×C10).236C24, (C4×C20).196C22, (C2×C20).188C23, D10.13D4⋊34C2, C2.57(D4⋊8D10), Dic5.Q8⋊33C2, (C2×D20).170C22, C4⋊Dic5.314C22, C22.257(C23×D5), D10⋊C4.10C22, C5⋊8(C22.33C24), (C4×Dic5).151C22, (C2×Dic5).268C23, C10.D4.52C22, (C22×D5).102C23, C2.29(Q8.10D10), (C2×Dic10).186C22, (D5×C4⋊C4)⋊36C2, C2.87(D5×C4○D4), C4⋊C4⋊D5⋊34C2, (C5×C42.C2)⋊9C2, C10.198(C2×C4○D4), (C2×C4×D5).135C22, (C2×C4).80(C22×D5), (C5×C4⋊C4).191C22, SmallGroup(320,1364)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 830 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×10], C5, C2×C4 [×7], C2×C4 [×11], D4 [×5], Q8, C23 [×3], D5 [×4], C10 [×3], C42, C42, C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×8], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×5], C20 [×7], D10 [×2], D10 [×8], C2×C10, C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2, C42.C2, C42⋊2C2 [×2], Dic10, C4×D5 [×6], D20 [×5], C2×Dic5 [×5], C2×C20 [×7], C22×D5 [×3], C22.33C24, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×6], C2×Dic10, C2×C4×D5 [×5], C2×D20 [×3], C4×D20, C42⋊2D5, Dic5.Q8, D5×C4⋊C4, D20⋊8C4, D10.13D4 [×4], C4⋊2D20, D10⋊Q8 [×2], D10⋊2Q8, C4⋊C4⋊D5, C5×C42.C2, C42.150D10
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], C22.33C24, C23×D5, Q8.10D10, D5×C4○D4, D4⋊8D10, C42.150D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 67 136 28)(2 78 137 39)(3 69 138 30)(4 80 139 21)(5 71 140 32)(6 62 121 23)(7 73 122 34)(8 64 123 25)(9 75 124 36)(10 66 125 27)(11 77 126 38)(12 68 127 29)(13 79 128 40)(14 70 129 31)(15 61 130 22)(16 72 131 33)(17 63 132 24)(18 74 133 35)(19 65 134 26)(20 76 135 37)(41 89 160 104)(42 100 141 115)(43 91 142 106)(44 82 143 117)(45 93 144 108)(46 84 145 119)(47 95 146 110)(48 86 147 101)(49 97 148 112)(50 88 149 103)(51 99 150 114)(52 90 151 105)(53 81 152 116)(54 92 153 107)(55 83 154 118)(56 94 155 109)(57 85 156 120)(58 96 157 111)(59 87 158 102)(60 98 159 113)
(1 100 126 105)(2 106 127 81)(3 82 128 107)(4 108 129 83)(5 84 130 109)(6 110 131 85)(7 86 132 111)(8 112 133 87)(9 88 134 113)(10 114 135 89)(11 90 136 115)(12 116 137 91)(13 92 138 117)(14 118 139 93)(15 94 140 119)(16 120 121 95)(17 96 122 101)(18 102 123 97)(19 98 124 103)(20 104 125 99)(21 144 70 55)(22 56 71 145)(23 146 72 57)(24 58 73 147)(25 148 74 59)(26 60 75 149)(27 150 76 41)(28 42 77 151)(29 152 78 43)(30 44 79 153)(31 154 80 45)(32 46 61 155)(33 156 62 47)(34 48 63 157)(35 158 64 49)(36 50 65 159)(37 160 66 51)(38 52 67 141)(39 142 68 53)(40 54 69 143)
(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)
(1 160 11 150)(2 149 12 159)(3 158 13 148)(4 147 14 157)(5 156 15 146)(6 145 16 155)(7 154 17 144)(8 143 18 153)(9 152 19 142)(10 141 20 151)(21 96 31 86)(22 85 32 95)(23 94 33 84)(24 83 34 93)(25 92 35 82)(26 81 36 91)(27 90 37 100)(28 99 38 89)(29 88 39 98)(30 97 40 87)(41 126 51 136)(42 135 52 125)(43 124 53 134)(44 133 54 123)(45 122 55 132)(46 131 56 121)(47 140 57 130)(48 129 58 139)(49 138 59 128)(50 127 60 137)(61 120 71 110)(62 109 72 119)(63 118 73 108)(64 107 74 117)(65 116 75 106)(66 105 76 115)(67 114 77 104)(68 103 78 113)(69 112 79 102)(70 101 80 111)
G:=sub<Sym(160)| (1,67,136,28)(2,78,137,39)(3,69,138,30)(4,80,139,21)(5,71,140,32)(6,62,121,23)(7,73,122,34)(8,64,123,25)(9,75,124,36)(10,66,125,27)(11,77,126,38)(12,68,127,29)(13,79,128,40)(14,70,129,31)(15,61,130,22)(16,72,131,33)(17,63,132,24)(18,74,133,35)(19,65,134,26)(20,76,135,37)(41,89,160,104)(42,100,141,115)(43,91,142,106)(44,82,143,117)(45,93,144,108)(46,84,145,119)(47,95,146,110)(48,86,147,101)(49,97,148,112)(50,88,149,103)(51,99,150,114)(52,90,151,105)(53,81,152,116)(54,92,153,107)(55,83,154,118)(56,94,155,109)(57,85,156,120)(58,96,157,111)(59,87,158,102)(60,98,159,113), (1,100,126,105)(2,106,127,81)(3,82,128,107)(4,108,129,83)(5,84,130,109)(6,110,131,85)(7,86,132,111)(8,112,133,87)(9,88,134,113)(10,114,135,89)(11,90,136,115)(12,116,137,91)(13,92,138,117)(14,118,139,93)(15,94,140,119)(16,120,121,95)(17,96,122,101)(18,102,123,97)(19,98,124,103)(20,104,125,99)(21,144,70,55)(22,56,71,145)(23,146,72,57)(24,58,73,147)(25,148,74,59)(26,60,75,149)(27,150,76,41)(28,42,77,151)(29,152,78,43)(30,44,79,153)(31,154,80,45)(32,46,61,155)(33,156,62,47)(34,48,63,157)(35,158,64,49)(36,50,65,159)(37,160,66,51)(38,52,67,141)(39,142,68,53)(40,54,69,143), (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), (1,160,11,150)(2,149,12,159)(3,158,13,148)(4,147,14,157)(5,156,15,146)(6,145,16,155)(7,154,17,144)(8,143,18,153)(9,152,19,142)(10,141,20,151)(21,96,31,86)(22,85,32,95)(23,94,33,84)(24,83,34,93)(25,92,35,82)(26,81,36,91)(27,90,37,100)(28,99,38,89)(29,88,39,98)(30,97,40,87)(41,126,51,136)(42,135,52,125)(43,124,53,134)(44,133,54,123)(45,122,55,132)(46,131,56,121)(47,140,57,130)(48,129,58,139)(49,138,59,128)(50,127,60,137)(61,120,71,110)(62,109,72,119)(63,118,73,108)(64,107,74,117)(65,116,75,106)(66,105,76,115)(67,114,77,104)(68,103,78,113)(69,112,79,102)(70,101,80,111)>;
G:=Group( (1,67,136,28)(2,78,137,39)(3,69,138,30)(4,80,139,21)(5,71,140,32)(6,62,121,23)(7,73,122,34)(8,64,123,25)(9,75,124,36)(10,66,125,27)(11,77,126,38)(12,68,127,29)(13,79,128,40)(14,70,129,31)(15,61,130,22)(16,72,131,33)(17,63,132,24)(18,74,133,35)(19,65,134,26)(20,76,135,37)(41,89,160,104)(42,100,141,115)(43,91,142,106)(44,82,143,117)(45,93,144,108)(46,84,145,119)(47,95,146,110)(48,86,147,101)(49,97,148,112)(50,88,149,103)(51,99,150,114)(52,90,151,105)(53,81,152,116)(54,92,153,107)(55,83,154,118)(56,94,155,109)(57,85,156,120)(58,96,157,111)(59,87,158,102)(60,98,159,113), (1,100,126,105)(2,106,127,81)(3,82,128,107)(4,108,129,83)(5,84,130,109)(6,110,131,85)(7,86,132,111)(8,112,133,87)(9,88,134,113)(10,114,135,89)(11,90,136,115)(12,116,137,91)(13,92,138,117)(14,118,139,93)(15,94,140,119)(16,120,121,95)(17,96,122,101)(18,102,123,97)(19,98,124,103)(20,104,125,99)(21,144,70,55)(22,56,71,145)(23,146,72,57)(24,58,73,147)(25,148,74,59)(26,60,75,149)(27,150,76,41)(28,42,77,151)(29,152,78,43)(30,44,79,153)(31,154,80,45)(32,46,61,155)(33,156,62,47)(34,48,63,157)(35,158,64,49)(36,50,65,159)(37,160,66,51)(38,52,67,141)(39,142,68,53)(40,54,69,143), (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), (1,160,11,150)(2,149,12,159)(3,158,13,148)(4,147,14,157)(5,156,15,146)(6,145,16,155)(7,154,17,144)(8,143,18,153)(9,152,19,142)(10,141,20,151)(21,96,31,86)(22,85,32,95)(23,94,33,84)(24,83,34,93)(25,92,35,82)(26,81,36,91)(27,90,37,100)(28,99,38,89)(29,88,39,98)(30,97,40,87)(41,126,51,136)(42,135,52,125)(43,124,53,134)(44,133,54,123)(45,122,55,132)(46,131,56,121)(47,140,57,130)(48,129,58,139)(49,138,59,128)(50,127,60,137)(61,120,71,110)(62,109,72,119)(63,118,73,108)(64,107,74,117)(65,116,75,106)(66,105,76,115)(67,114,77,104)(68,103,78,113)(69,112,79,102)(70,101,80,111) );
G=PermutationGroup([(1,67,136,28),(2,78,137,39),(3,69,138,30),(4,80,139,21),(5,71,140,32),(6,62,121,23),(7,73,122,34),(8,64,123,25),(9,75,124,36),(10,66,125,27),(11,77,126,38),(12,68,127,29),(13,79,128,40),(14,70,129,31),(15,61,130,22),(16,72,131,33),(17,63,132,24),(18,74,133,35),(19,65,134,26),(20,76,135,37),(41,89,160,104),(42,100,141,115),(43,91,142,106),(44,82,143,117),(45,93,144,108),(46,84,145,119),(47,95,146,110),(48,86,147,101),(49,97,148,112),(50,88,149,103),(51,99,150,114),(52,90,151,105),(53,81,152,116),(54,92,153,107),(55,83,154,118),(56,94,155,109),(57,85,156,120),(58,96,157,111),(59,87,158,102),(60,98,159,113)], [(1,100,126,105),(2,106,127,81),(3,82,128,107),(4,108,129,83),(5,84,130,109),(6,110,131,85),(7,86,132,111),(8,112,133,87),(9,88,134,113),(10,114,135,89),(11,90,136,115),(12,116,137,91),(13,92,138,117),(14,118,139,93),(15,94,140,119),(16,120,121,95),(17,96,122,101),(18,102,123,97),(19,98,124,103),(20,104,125,99),(21,144,70,55),(22,56,71,145),(23,146,72,57),(24,58,73,147),(25,148,74,59),(26,60,75,149),(27,150,76,41),(28,42,77,151),(29,152,78,43),(30,44,79,153),(31,154,80,45),(32,46,61,155),(33,156,62,47),(34,48,63,157),(35,158,64,49),(36,50,65,159),(37,160,66,51),(38,52,67,141),(39,142,68,53),(40,54,69,143)], [(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)], [(1,160,11,150),(2,149,12,159),(3,158,13,148),(4,147,14,157),(5,156,15,146),(6,145,16,155),(7,154,17,144),(8,143,18,153),(9,152,19,142),(10,141,20,151),(21,96,31,86),(22,85,32,95),(23,94,33,84),(24,83,34,93),(25,92,35,82),(26,81,36,91),(27,90,37,100),(28,99,38,89),(29,88,39,98),(30,97,40,87),(41,126,51,136),(42,135,52,125),(43,124,53,134),(44,133,54,123),(45,122,55,132),(46,131,56,121),(47,140,57,130),(48,129,58,139),(49,138,59,128),(50,127,60,137),(61,120,71,110),(62,109,72,119),(63,118,73,108),(64,107,74,117),(65,116,75,106),(66,105,76,115),(67,114,77,104),(68,103,78,113),(69,112,79,102),(70,101,80,111)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 23 | 9 | 2 |
| 0 | 0 | 0 | 0 | 0 | 39 | 0 | 32 |
| 1 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 9 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 32 | 39 |
| 0 | 0 | 0 | 0 | 0 | 9 | 32 | 40 |
| 0 | 0 | 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 | 1 | 32 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,37,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,18,9,23,39,0,0,0,0,0,0,9,0,0,0,0,0,0,0,2,32],[1,7,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,31,35,0,0,0,0,0,0,37,10,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,18,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,40],[40,34,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,31,35,0,0,0,0,0,0,37,10,0,0,0,0,0,0,0,0,0,0,9,1,0,0,0,0,18,9,23,39,0,0,0,0,32,32,0,1,0,0,0,0,39,40,0,32] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2+ (1+4) | 2- (1+4) | Q8.10D10 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.150D10 | C4×D20 | C42⋊2D5 | Dic5.Q8 | D5×C4⋊C4 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | D10 | C42 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 12 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{150}D_{10} % in TeX
G:=Group("C4^2.150D10"); // GroupNames label
G:=SmallGroup(320,1364);
// by ID
G=gap.SmallGroup(320,1364);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀