metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.95D10, C10.972+ (1+4), (C4×D20)⋊10C2, D10⋊D4⋊3C2, C20⋊7D4⋊29C2, C4⋊2D20⋊12C2, C4⋊C4.272D10, C42⋊2D5⋊4C2, D20⋊8C4⋊12C2, C4.96(C4○D20), C42⋊C2⋊14D5, (C4×C20).25C22, (C2×C10).74C24, C22⋊C4.98D10, C4.Dic10⋊13C2, D10.51(C4○D4), D10.12D4⋊4C2, C20.198(C4○D4), C2.9(D4⋊8D10), (C2×C20).149C23, (C22×C4).195D10, C23.86(C22×D5), (C2×D20).143C22, D10⋊C4.3C22, C4⋊Dic5.195C22, (C4×Dic5).78C22, (C2×Dic5).27C23, (C22×D5).22C23, C22.103(C23×D5), C23.D5.97C22, (C22×C10).144C23, (C22×C20).232C22, C5⋊2(C22.47C24), C10.D4.98C22, (D5×C4⋊C4)⋊13C2, (C4×C5⋊D4)⋊12C2, C2.13(D5×C4○D4), C2.33(C2×C4○D20), (C2×C4×D5).70C22, C10.134(C2×C4○D4), (C5×C42⋊C2)⋊16C2, (C5×C4⋊C4).310C22, (C2×C4).276(C22×D5), (C2×C5⋊D4).112C22, (C5×C22⋊C4).114C22, SmallGroup(320,1202)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 902 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×10], C23, C23 [×3], D5 [×4], C10 [×3], C10, C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×5], C2×D4 [×6], Dic5 [×5], C20 [×2], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D5 [×6], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×D5 [×2], C22×C10, C22.47C24, C4×Dic5, C10.D4, C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×6], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, C4×D20 [×2], C42⋊2D5 [×2], D10.12D4 [×2], D10⋊D4 [×2], C4.Dic10, D5×C4⋊C4, D20⋊8C4, C4⋊2D20, C4×C5⋊D4, C20⋊7D4, C5×C42⋊C2, C42.95D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.47C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4⋊8D10, C42.95D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 109 134 83)(2 120 135 94)(3 111 136 85)(4 102 137 96)(5 113 138 87)(6 104 139 98)(7 115 140 89)(8 106 121 100)(9 117 122 91)(10 108 123 82)(11 119 124 93)(12 110 125 84)(13 101 126 95)(14 112 127 86)(15 103 128 97)(16 114 129 88)(17 105 130 99)(18 116 131 90)(19 107 132 81)(20 118 133 92)(21 149 61 43)(22 160 62 54)(23 151 63 45)(24 142 64 56)(25 153 65 47)(26 144 66 58)(27 155 67 49)(28 146 68 60)(29 157 69 51)(30 148 70 42)(31 159 71 53)(32 150 72 44)(33 141 73 55)(34 152 74 46)(35 143 75 57)(36 154 76 48)(37 145 77 59)(38 156 78 50)(39 147 79 41)(40 158 80 52)
(1 143 11 153)(2 144 12 154)(3 145 13 155)(4 146 14 156)(5 147 15 157)(6 148 16 158)(7 149 17 159)(8 150 18 160)(9 151 19 141)(10 152 20 142)(21 99 31 89)(22 100 32 90)(23 81 33 91)(24 82 34 92)(25 83 35 93)(26 84 36 94)(27 85 37 95)(28 86 38 96)(29 87 39 97)(30 88 40 98)(41 128 51 138)(42 129 52 139)(43 130 53 140)(44 131 54 121)(45 132 55 122)(46 133 56 123)(47 134 57 124)(48 135 58 125)(49 136 59 126)(50 137 60 127)(61 105 71 115)(62 106 72 116)(63 107 73 117)(64 108 74 118)(65 109 75 119)(66 110 76 120)(67 111 77 101)(68 112 78 102)(69 113 79 103)(70 114 80 104)
(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 34 11 24)(2 23 12 33)(3 32 13 22)(4 21 14 31)(5 30 15 40)(6 39 16 29)(7 28 17 38)(8 37 18 27)(9 26 19 36)(10 35 20 25)(41 98 51 88)(42 87 52 97)(43 96 53 86)(44 85 54 95)(45 94 55 84)(46 83 56 93)(47 92 57 82)(48 81 58 91)(49 90 59 100)(50 99 60 89)(61 127 71 137)(62 136 72 126)(63 125 73 135)(64 134 74 124)(65 123 75 133)(66 132 76 122)(67 121 77 131)(68 130 78 140)(69 139 79 129)(70 128 80 138)(101 150 111 160)(102 159 112 149)(103 148 113 158)(104 157 114 147)(105 146 115 156)(106 155 116 145)(107 144 117 154)(108 153 118 143)(109 142 119 152)(110 151 120 141)
G:=sub<Sym(160)| (1,109,134,83)(2,120,135,94)(3,111,136,85)(4,102,137,96)(5,113,138,87)(6,104,139,98)(7,115,140,89)(8,106,121,100)(9,117,122,91)(10,108,123,82)(11,119,124,93)(12,110,125,84)(13,101,126,95)(14,112,127,86)(15,103,128,97)(16,114,129,88)(17,105,130,99)(18,116,131,90)(19,107,132,81)(20,118,133,92)(21,149,61,43)(22,160,62,54)(23,151,63,45)(24,142,64,56)(25,153,65,47)(26,144,66,58)(27,155,67,49)(28,146,68,60)(29,157,69,51)(30,148,70,42)(31,159,71,53)(32,150,72,44)(33,141,73,55)(34,152,74,46)(35,143,75,57)(36,154,76,48)(37,145,77,59)(38,156,78,50)(39,147,79,41)(40,158,80,52), (1,143,11,153)(2,144,12,154)(3,145,13,155)(4,146,14,156)(5,147,15,157)(6,148,16,158)(7,149,17,159)(8,150,18,160)(9,151,19,141)(10,152,20,142)(21,99,31,89)(22,100,32,90)(23,81,33,91)(24,82,34,92)(25,83,35,93)(26,84,36,94)(27,85,37,95)(28,86,38,96)(29,87,39,97)(30,88,40,98)(41,128,51,138)(42,129,52,139)(43,130,53,140)(44,131,54,121)(45,132,55,122)(46,133,56,123)(47,134,57,124)(48,135,58,125)(49,136,59,126)(50,137,60,127)(61,105,71,115)(62,106,72,116)(63,107,73,117)(64,108,74,118)(65,109,75,119)(66,110,76,120)(67,111,77,101)(68,112,78,102)(69,113,79,103)(70,114,80,104), (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), (1,34,11,24)(2,23,12,33)(3,32,13,22)(4,21,14,31)(5,30,15,40)(6,39,16,29)(7,28,17,38)(8,37,18,27)(9,26,19,36)(10,35,20,25)(41,98,51,88)(42,87,52,97)(43,96,53,86)(44,85,54,95)(45,94,55,84)(46,83,56,93)(47,92,57,82)(48,81,58,91)(49,90,59,100)(50,99,60,89)(61,127,71,137)(62,136,72,126)(63,125,73,135)(64,134,74,124)(65,123,75,133)(66,132,76,122)(67,121,77,131)(68,130,78,140)(69,139,79,129)(70,128,80,138)(101,150,111,160)(102,159,112,149)(103,148,113,158)(104,157,114,147)(105,146,115,156)(106,155,116,145)(107,144,117,154)(108,153,118,143)(109,142,119,152)(110,151,120,141)>;
G:=Group( (1,109,134,83)(2,120,135,94)(3,111,136,85)(4,102,137,96)(5,113,138,87)(6,104,139,98)(7,115,140,89)(8,106,121,100)(9,117,122,91)(10,108,123,82)(11,119,124,93)(12,110,125,84)(13,101,126,95)(14,112,127,86)(15,103,128,97)(16,114,129,88)(17,105,130,99)(18,116,131,90)(19,107,132,81)(20,118,133,92)(21,149,61,43)(22,160,62,54)(23,151,63,45)(24,142,64,56)(25,153,65,47)(26,144,66,58)(27,155,67,49)(28,146,68,60)(29,157,69,51)(30,148,70,42)(31,159,71,53)(32,150,72,44)(33,141,73,55)(34,152,74,46)(35,143,75,57)(36,154,76,48)(37,145,77,59)(38,156,78,50)(39,147,79,41)(40,158,80,52), (1,143,11,153)(2,144,12,154)(3,145,13,155)(4,146,14,156)(5,147,15,157)(6,148,16,158)(7,149,17,159)(8,150,18,160)(9,151,19,141)(10,152,20,142)(21,99,31,89)(22,100,32,90)(23,81,33,91)(24,82,34,92)(25,83,35,93)(26,84,36,94)(27,85,37,95)(28,86,38,96)(29,87,39,97)(30,88,40,98)(41,128,51,138)(42,129,52,139)(43,130,53,140)(44,131,54,121)(45,132,55,122)(46,133,56,123)(47,134,57,124)(48,135,58,125)(49,136,59,126)(50,137,60,127)(61,105,71,115)(62,106,72,116)(63,107,73,117)(64,108,74,118)(65,109,75,119)(66,110,76,120)(67,111,77,101)(68,112,78,102)(69,113,79,103)(70,114,80,104), (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), (1,34,11,24)(2,23,12,33)(3,32,13,22)(4,21,14,31)(5,30,15,40)(6,39,16,29)(7,28,17,38)(8,37,18,27)(9,26,19,36)(10,35,20,25)(41,98,51,88)(42,87,52,97)(43,96,53,86)(44,85,54,95)(45,94,55,84)(46,83,56,93)(47,92,57,82)(48,81,58,91)(49,90,59,100)(50,99,60,89)(61,127,71,137)(62,136,72,126)(63,125,73,135)(64,134,74,124)(65,123,75,133)(66,132,76,122)(67,121,77,131)(68,130,78,140)(69,139,79,129)(70,128,80,138)(101,150,111,160)(102,159,112,149)(103,148,113,158)(104,157,114,147)(105,146,115,156)(106,155,116,145)(107,144,117,154)(108,153,118,143)(109,142,119,152)(110,151,120,141) );
G=PermutationGroup([(1,109,134,83),(2,120,135,94),(3,111,136,85),(4,102,137,96),(5,113,138,87),(6,104,139,98),(7,115,140,89),(8,106,121,100),(9,117,122,91),(10,108,123,82),(11,119,124,93),(12,110,125,84),(13,101,126,95),(14,112,127,86),(15,103,128,97),(16,114,129,88),(17,105,130,99),(18,116,131,90),(19,107,132,81),(20,118,133,92),(21,149,61,43),(22,160,62,54),(23,151,63,45),(24,142,64,56),(25,153,65,47),(26,144,66,58),(27,155,67,49),(28,146,68,60),(29,157,69,51),(30,148,70,42),(31,159,71,53),(32,150,72,44),(33,141,73,55),(34,152,74,46),(35,143,75,57),(36,154,76,48),(37,145,77,59),(38,156,78,50),(39,147,79,41),(40,158,80,52)], [(1,143,11,153),(2,144,12,154),(3,145,13,155),(4,146,14,156),(5,147,15,157),(6,148,16,158),(7,149,17,159),(8,150,18,160),(9,151,19,141),(10,152,20,142),(21,99,31,89),(22,100,32,90),(23,81,33,91),(24,82,34,92),(25,83,35,93),(26,84,36,94),(27,85,37,95),(28,86,38,96),(29,87,39,97),(30,88,40,98),(41,128,51,138),(42,129,52,139),(43,130,53,140),(44,131,54,121),(45,132,55,122),(46,133,56,123),(47,134,57,124),(48,135,58,125),(49,136,59,126),(50,137,60,127),(61,105,71,115),(62,106,72,116),(63,107,73,117),(64,108,74,118),(65,109,75,119),(66,110,76,120),(67,111,77,101),(68,112,78,102),(69,113,79,103),(70,114,80,104)], [(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)], [(1,34,11,24),(2,23,12,33),(3,32,13,22),(4,21,14,31),(5,30,15,40),(6,39,16,29),(7,28,17,38),(8,37,18,27),(9,26,19,36),(10,35,20,25),(41,98,51,88),(42,87,52,97),(43,96,53,86),(44,85,54,95),(45,94,55,84),(46,83,56,93),(47,92,57,82),(48,81,58,91),(49,90,59,100),(50,99,60,89),(61,127,71,137),(62,136,72,126),(63,125,73,135),(64,134,74,124),(65,123,75,133),(66,132,76,122),(67,121,77,131),(68,130,78,140),(69,139,79,129),(70,128,80,138),(101,150,111,160),(102,159,112,149),(103,148,113,158),(104,157,114,147),(105,146,115,156),(106,155,116,145),(107,144,117,154),(108,153,118,143),(109,142,119,152),(110,151,120,141)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 20 | 9 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 36 |
| 0 | 0 | 25 | 1 |
| 14 | 27 | 0 | 0 |
| 11 | 27 | 0 | 0 |
| 0 | 0 | 40 | 36 |
| 0 | 0 | 25 | 1 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,32,20,0,0,0,9],[24,40,0,0,1,17,0,0,0,0,9,0,0,0,0,9],[34,7,0,0,34,1,0,0,0,0,40,25,0,0,36,1],[14,11,0,0,27,27,0,0,0,0,40,25,0,0,36,1] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.95D10 | C4×D20 | C42⋊2D5 | D10.12D4 | D10⋊D4 | C4.Dic10 | D5×C4⋊C4 | D20⋊8C4 | C4⋊2D20 | C4×C5⋊D4 | C20⋊7D4 | C5×C42⋊C2 | C42⋊C2 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{95}D_{10} % in TeX
G:=Group("C4^2.95D10"); // GroupNames label
G:=SmallGroup(320,1202);
// by ID
G=gap.SmallGroup(320,1202);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀