metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.154D10, C10.302- (1+4), C20⋊Q8⋊37C2, C4⋊C4.210D10, C42.C2⋊10D5, (C4×Dic10)⋊49C2, D10⋊Q8.3C2, C4.Dic10⋊36C2, C42⋊2D5.1C2, Dic5⋊3Q8⋊37C2, (C2×C20).602C23, (C4×C20).199C22, (C2×C10).240C24, Dic5.19(C4○D4), Dic5.Q8⋊35C2, C4⋊Dic5.316C22, C22.261(C23×D5), D10⋊C4.42C22, C5⋊4(C22.35C24), (C2×Dic5).270C23, (C4×Dic5).236C22, (C22×D5).105C23, C2.59(D4.10D10), C2.31(Q8.10D10), (C2×Dic10).260C22, C10.D4.162C22, C2.91(D5×C4○D4), C4⋊C4⋊D5.2C2, C4⋊C4⋊7D5.13C2, C10.202(C2×C4○D4), (C5×C42.C2)⋊13C2, (C2×C4×D5).139C22, (C5×C4⋊C4).195C22, (C2×C4).205(C22×D5), SmallGroup(320,1368)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 590 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2, C4 [×15], C22, C22 [×3], C5, C2×C4 [×7], C2×C4 [×9], Q8 [×4], C23, D5, C10 [×3], C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×14], C22×C4, C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×7], D10 [×3], C2×C10, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×2], C2×Dic5 [×7], C2×C20 [×7], C22×D5, C22.35C24, C4×Dic5 [×5], C10.D4 [×10], C4⋊Dic5 [×4], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×2], C2×C4×D5, C4×Dic10, C42⋊2D5, Dic5⋊3Q8, C20⋊Q8, Dic5.Q8 [×3], C4.Dic10, C4⋊C4⋊7D5, D10⋊Q8 [×2], C4⋊C4⋊D5 [×3], C5×C42.C2, C42.154D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2- (1+4) [×2], C22×D5 [×7], C22.35C24, C23×D5, Q8.10D10, D5×C4○D4, D4.10D10, C42.154D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 146 11 156)(2 111 12 101)(3 148 13 158)(4 113 14 103)(5 150 15 160)(6 115 16 105)(7 152 17 142)(8 117 18 107)(9 154 19 144)(10 119 20 109)(21 151 31 141)(22 116 32 106)(23 153 33 143)(24 118 34 108)(25 155 35 145)(26 120 36 110)(27 157 37 147)(28 102 38 112)(29 159 39 149)(30 104 40 114)(41 69 51 79)(42 136 52 126)(43 71 53 61)(44 138 54 128)(45 73 55 63)(46 140 56 130)(47 75 57 65)(48 122 58 132)(49 77 59 67)(50 124 60 134)(62 81 72 91)(64 83 74 93)(66 85 76 95)(68 87 78 97)(70 89 80 99)(82 139 92 129)(84 121 94 131)(86 123 96 133)(88 125 98 135)(90 127 100 137)
(1 52 26 99)(2 100 27 53)(3 54 28 81)(4 82 29 55)(5 56 30 83)(6 84 31 57)(7 58 32 85)(8 86 33 59)(9 60 34 87)(10 88 35 41)(11 42 36 89)(12 90 37 43)(13 44 38 91)(14 92 39 45)(15 46 40 93)(16 94 21 47)(17 48 22 95)(18 96 23 49)(19 50 24 97)(20 98 25 51)(61 111 137 157)(62 158 138 112)(63 113 139 159)(64 160 140 114)(65 115 121 141)(66 142 122 116)(67 117 123 143)(68 144 124 118)(69 119 125 145)(70 146 126 120)(71 101 127 147)(72 148 128 102)(73 103 129 149)(74 150 130 104)(75 105 131 151)(76 152 132 106)(77 107 133 153)(78 154 134 108)(79 109 135 155)(80 156 136 110)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 30 31 40)(22 39 32 29)(23 28 33 38)(24 37 34 27)(25 26 35 36)(41 99 51 89)(42 88 52 98)(43 97 53 87)(44 86 54 96)(45 95 55 85)(46 84 56 94)(47 93 57 83)(48 82 58 92)(49 91 59 81)(50 100 60 90)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)(101 144 111 154)(102 153 112 143)(103 142 113 152)(104 151 114 141)(105 160 115 150)(106 149 116 159)(107 158 117 148)(108 147 118 157)(109 156 119 146)(110 145 120 155)(121 140 131 130)(122 129 132 139)(123 138 133 128)(124 127 134 137)(125 136 135 126)
G:=sub<Sym(160)| (1,146,11,156)(2,111,12,101)(3,148,13,158)(4,113,14,103)(5,150,15,160)(6,115,16,105)(7,152,17,142)(8,117,18,107)(9,154,19,144)(10,119,20,109)(21,151,31,141)(22,116,32,106)(23,153,33,143)(24,118,34,108)(25,155,35,145)(26,120,36,110)(27,157,37,147)(28,102,38,112)(29,159,39,149)(30,104,40,114)(41,69,51,79)(42,136,52,126)(43,71,53,61)(44,138,54,128)(45,73,55,63)(46,140,56,130)(47,75,57,65)(48,122,58,132)(49,77,59,67)(50,124,60,134)(62,81,72,91)(64,83,74,93)(66,85,76,95)(68,87,78,97)(70,89,80,99)(82,139,92,129)(84,121,94,131)(86,123,96,133)(88,125,98,135)(90,127,100,137), (1,52,26,99)(2,100,27,53)(3,54,28,81)(4,82,29,55)(5,56,30,83)(6,84,31,57)(7,58,32,85)(8,86,33,59)(9,60,34,87)(10,88,35,41)(11,42,36,89)(12,90,37,43)(13,44,38,91)(14,92,39,45)(15,46,40,93)(16,94,21,47)(17,48,22,95)(18,96,23,49)(19,50,24,97)(20,98,25,51)(61,111,137,157)(62,158,138,112)(63,113,139,159)(64,160,140,114)(65,115,121,141)(66,142,122,116)(67,117,123,143)(68,144,124,118)(69,119,125,145)(70,146,126,120)(71,101,127,147)(72,148,128,102)(73,103,129,149)(74,150,130,104)(75,105,131,151)(76,152,132,106)(77,107,133,153)(78,154,134,108)(79,109,135,155)(80,156,136,110), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,30,31,40)(22,39,32,29)(23,28,33,38)(24,37,34,27)(25,26,35,36)(41,99,51,89)(42,88,52,98)(43,97,53,87)(44,86,54,96)(45,95,55,85)(46,84,56,94)(47,93,57,83)(48,82,58,92)(49,91,59,81)(50,100,60,90)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(101,144,111,154)(102,153,112,143)(103,142,113,152)(104,151,114,141)(105,160,115,150)(106,149,116,159)(107,158,117,148)(108,147,118,157)(109,156,119,146)(110,145,120,155)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126)>;
G:=Group( (1,146,11,156)(2,111,12,101)(3,148,13,158)(4,113,14,103)(5,150,15,160)(6,115,16,105)(7,152,17,142)(8,117,18,107)(9,154,19,144)(10,119,20,109)(21,151,31,141)(22,116,32,106)(23,153,33,143)(24,118,34,108)(25,155,35,145)(26,120,36,110)(27,157,37,147)(28,102,38,112)(29,159,39,149)(30,104,40,114)(41,69,51,79)(42,136,52,126)(43,71,53,61)(44,138,54,128)(45,73,55,63)(46,140,56,130)(47,75,57,65)(48,122,58,132)(49,77,59,67)(50,124,60,134)(62,81,72,91)(64,83,74,93)(66,85,76,95)(68,87,78,97)(70,89,80,99)(82,139,92,129)(84,121,94,131)(86,123,96,133)(88,125,98,135)(90,127,100,137), (1,52,26,99)(2,100,27,53)(3,54,28,81)(4,82,29,55)(5,56,30,83)(6,84,31,57)(7,58,32,85)(8,86,33,59)(9,60,34,87)(10,88,35,41)(11,42,36,89)(12,90,37,43)(13,44,38,91)(14,92,39,45)(15,46,40,93)(16,94,21,47)(17,48,22,95)(18,96,23,49)(19,50,24,97)(20,98,25,51)(61,111,137,157)(62,158,138,112)(63,113,139,159)(64,160,140,114)(65,115,121,141)(66,142,122,116)(67,117,123,143)(68,144,124,118)(69,119,125,145)(70,146,126,120)(71,101,127,147)(72,148,128,102)(73,103,129,149)(74,150,130,104)(75,105,131,151)(76,152,132,106)(77,107,133,153)(78,154,134,108)(79,109,135,155)(80,156,136,110), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,30,31,40)(22,39,32,29)(23,28,33,38)(24,37,34,27)(25,26,35,36)(41,99,51,89)(42,88,52,98)(43,97,53,87)(44,86,54,96)(45,95,55,85)(46,84,56,94)(47,93,57,83)(48,82,58,92)(49,91,59,81)(50,100,60,90)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(101,144,111,154)(102,153,112,143)(103,142,113,152)(104,151,114,141)(105,160,115,150)(106,149,116,159)(107,158,117,148)(108,147,118,157)(109,156,119,146)(110,145,120,155)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126) );
G=PermutationGroup([(1,146,11,156),(2,111,12,101),(3,148,13,158),(4,113,14,103),(5,150,15,160),(6,115,16,105),(7,152,17,142),(8,117,18,107),(9,154,19,144),(10,119,20,109),(21,151,31,141),(22,116,32,106),(23,153,33,143),(24,118,34,108),(25,155,35,145),(26,120,36,110),(27,157,37,147),(28,102,38,112),(29,159,39,149),(30,104,40,114),(41,69,51,79),(42,136,52,126),(43,71,53,61),(44,138,54,128),(45,73,55,63),(46,140,56,130),(47,75,57,65),(48,122,58,132),(49,77,59,67),(50,124,60,134),(62,81,72,91),(64,83,74,93),(66,85,76,95),(68,87,78,97),(70,89,80,99),(82,139,92,129),(84,121,94,131),(86,123,96,133),(88,125,98,135),(90,127,100,137)], [(1,52,26,99),(2,100,27,53),(3,54,28,81),(4,82,29,55),(5,56,30,83),(6,84,31,57),(7,58,32,85),(8,86,33,59),(9,60,34,87),(10,88,35,41),(11,42,36,89),(12,90,37,43),(13,44,38,91),(14,92,39,45),(15,46,40,93),(16,94,21,47),(17,48,22,95),(18,96,23,49),(19,50,24,97),(20,98,25,51),(61,111,137,157),(62,158,138,112),(63,113,139,159),(64,160,140,114),(65,115,121,141),(66,142,122,116),(67,117,123,143),(68,144,124,118),(69,119,125,145),(70,146,126,120),(71,101,127,147),(72,148,128,102),(73,103,129,149),(74,150,130,104),(75,105,131,151),(76,152,132,106),(77,107,133,153),(78,154,134,108),(79,109,135,155),(80,156,136,110)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,30,31,40),(22,39,32,29),(23,28,33,38),(24,37,34,27),(25,26,35,36),(41,99,51,89),(42,88,52,98),(43,97,53,87),(44,86,54,96),(45,95,55,85),(46,84,56,94),(47,93,57,83),(48,82,58,92),(49,91,59,81),(50,100,60,90),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70),(101,144,111,154),(102,153,112,143),(103,142,113,152),(104,151,114,141),(105,160,115,150),(106,149,116,159),(107,158,117,148),(108,147,118,157),(109,156,119,146),(110,145,120,155),(121,140,131,130),(122,129,132,139),(123,138,133,128),(124,127,134,137),(125,136,135,126)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 9 | 0 | 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 23 | 0 | 0 | 0 | 0 |
| 9 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 25 | 1 |
| 0 | 0 | 0 | 0 | 3 | 19 | 23 | 22 |
| 0 | 0 | 0 | 0 | 7 | 2 | 1 | 34 |
| 0 | 0 | 0 | 0 | 40 | 1 | 26 | 18 |
| 1 | 0 | 39 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 39 | 0 | 0 | 0 | 0 |
| 1 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 | 22 | 21 |
| 0 | 0 | 0 | 0 | 1 | 24 | 12 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 23 |
| 1 | 29 | 28 | 13 | 0 | 0 | 0 | 0 |
| 12 | 11 | 28 | 24 | 0 | 0 | 0 | 0 |
| 15 | 15 | 40 | 12 | 0 | 0 | 0 | 0 |
| 26 | 23 | 29 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 9 | 6 | 24 |
| 0 | 0 | 0 | 0 | 16 | 32 | 33 | 11 |
| 0 | 0 | 0 | 0 | 35 | 6 | 33 | 1 |
| 0 | 0 | 0 | 0 | 27 | 33 | 21 | 9 |
| 1 | 29 | 28 | 13 | 0 | 0 | 0 | 0 |
| 35 | 40 | 17 | 13 | 0 | 0 | 0 | 0 |
| 27 | 14 | 29 | 1 | 0 | 0 | 0 | 0 |
| 12 | 14 | 11 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 32 | 21 | 5 |
| 0 | 0 | 0 | 0 | 38 | 9 | 24 | 30 |
| 0 | 0 | 0 | 0 | 34 | 35 | 11 | 2 |
| 0 | 0 | 0 | 0 | 1 | 8 | 23 | 35 |
G:=sub<GL(8,GF(41))| [9,0,9,0,0,0,0,0,0,9,0,9,0,0,0,0,23,0,32,0,0,0,0,0,0,23,0,32,0,0,0,0,0,0,0,0,3,3,7,40,0,0,0,0,7,19,2,1,0,0,0,0,25,23,1,26,0,0,0,0,1,22,34,18],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,39,0,40,0,0,0,0,0,0,39,0,40,0,0,0,0,0,0,0,0,17,1,0,0,0,0,0,0,40,24,0,0,0,0,0,0,22,12,18,1,0,0,0,0,21,7,5,23],[1,12,15,26,0,0,0,0,29,11,15,23,0,0,0,0,28,28,40,29,0,0,0,0,13,24,12,30,0,0,0,0,0,0,0,0,8,16,35,27,0,0,0,0,9,32,6,33,0,0,0,0,6,33,33,21,0,0,0,0,24,11,1,9],[1,35,27,12,0,0,0,0,29,40,14,14,0,0,0,0,28,17,29,11,0,0,0,0,13,13,1,12,0,0,0,0,0,0,0,0,27,38,34,1,0,0,0,0,32,9,35,8,0,0,0,0,21,24,11,23,0,0,0,0,5,30,2,35] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 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 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2- (1+4) | Q8.10D10 | D5×C4○D4 | D4.10D10 |
| kernel | C42.154D10 | C4×Dic10 | C42⋊2D5 | Dic5⋊3Q8 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | C4⋊C4⋊7D5 | D10⋊Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | Dic5 | C42 | C4⋊C4 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 2 | 3 | 1 | 2 | 4 | 2 | 12 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{154}D_{10} % in TeX
G:=Group("C4^2.154D10"); // GroupNames label
G:=SmallGroup(320,1368);
// by ID
G=gap.SmallGroup(320,1368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,1571,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀