metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.141D10, C10.902- (1+4), C4.33(D4×D5), (C4×D5).12D4, C4.4D4⋊9D5, C20.62(C2×D4), C20⋊2Q8⋊30C2, D10.80(C2×D4), (C2×D4).172D10, C42⋊D5⋊20C2, (C2×C20).80C23, (C2×Q8).136D10, C22⋊C4.35D10, Dic5.91(C2×D4), C10.89(C22×D4), Dic5⋊Q8⋊24C2, C20.17D4⋊24C2, (C2×C10).219C24, (C4×C20).185C22, C4⋊Dic5.51C22, D10.12D4⋊41C2, C23.41(C22×D5), (D4×C10).154C22, (C22×C10).49C23, (Q8×C10).126C22, C22.240(C23×D5), Dic5.14D4⋊40C2, C23.D5.54C22, C5⋊4(C23.38C23), (C4×Dic5).141C22, (C2×Dic5).114C23, (C22×D5).224C23, C2.51(D4.10D10), D10⋊C4.110C22, (C2×Dic10).183C22, C10.D4.120C22, (C22×Dic5).142C22, (C2×Q8×D5)⋊10C2, C2.62(C2×D4×D5), (C5×C4.4D4)⋊11C2, (C2×C4×D5).129C22, (C2×D4⋊2D5).11C2, (C2×C4).194(C22×D5), (C2×C5⋊D4).59C22, (C5×C22⋊C4).64C22, SmallGroup(320,1347)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 926 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4, C2×C4 [×4], C2×C4 [×19], D4 [×6], Q8 [×10], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10 [×2], C23.38C23, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×4], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], D4⋊2D5 [×4], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C20⋊2Q8, C42⋊D5, Dic5.14D4 [×4], D10.12D4 [×4], C20.17D4, Dic5⋊Q8, C5×C4.4D4, C2×D4⋊2D5, C2×Q8×D5, C42.141D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- (1+4) [×2], C22×D5 [×7], C23.38C23, D4×D5 [×2], C23×D5, C2×D4×D5, D4.10D10 [×2], C42.141D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 21 11 31)(2 32 12 22)(3 23 13 33)(4 34 14 24)(5 25 15 35)(6 36 16 26)(7 27 17 37)(8 38 18 28)(9 29 19 39)(10 40 20 30)(41 143 51 153)(42 154 52 144)(43 145 53 155)(44 156 54 146)(45 147 55 157)(46 158 56 148)(47 149 57 159)(48 160 58 150)(49 151 59 141)(50 142 60 152)(61 131 71 121)(62 122 72 132)(63 133 73 123)(64 124 74 134)(65 135 75 125)(66 126 76 136)(67 137 77 127)(68 128 78 138)(69 139 79 129)(70 130 80 140)(81 116 91 106)(82 107 92 117)(83 118 93 108)(84 109 94 119)(85 120 95 110)(86 111 96 101)(87 102 97 112)(88 113 98 103)(89 104 99 114)(90 115 100 105)
(1 79 86 49)(2 60 87 70)(3 61 88 51)(4 42 89 72)(5 63 90 53)(6 44 91 74)(7 65 92 55)(8 46 93 76)(9 67 94 57)(10 48 95 78)(11 69 96 59)(12 50 97 80)(13 71 98 41)(14 52 99 62)(15 73 100 43)(16 54 81 64)(17 75 82 45)(18 56 83 66)(19 77 84 47)(20 58 85 68)(21 129 111 151)(22 142 112 140)(23 131 113 153)(24 144 114 122)(25 133 115 155)(26 146 116 124)(27 135 117 157)(28 148 118 126)(29 137 119 159)(30 150 120 128)(31 139 101 141)(32 152 102 130)(33 121 103 143)(34 154 104 132)(35 123 105 145)(36 156 106 134)(37 125 107 147)(38 158 108 136)(39 127 109 149)(40 160 110 138)
(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 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 76 71 66)(62 65 72 75)(63 74 73 64)(67 70 77 80)(68 79 78 69)(81 90 91 100)(82 99 92 89)(83 88 93 98)(84 97 94 87)(85 86 95 96)(101 110 111 120)(102 119 112 109)(103 108 113 118)(104 117 114 107)(105 106 115 116)(121 136 131 126)(122 125 132 135)(123 134 133 124)(127 130 137 140)(128 139 138 129)(141 160 151 150)(142 149 152 159)(143 158 153 148)(144 147 154 157)(145 156 155 146)
G:=sub<Sym(160)| (1,21,11,31)(2,32,12,22)(3,23,13,33)(4,34,14,24)(5,25,15,35)(6,36,16,26)(7,27,17,37)(8,38,18,28)(9,29,19,39)(10,40,20,30)(41,143,51,153)(42,154,52,144)(43,145,53,155)(44,156,54,146)(45,147,55,157)(46,158,56,148)(47,149,57,159)(48,160,58,150)(49,151,59,141)(50,142,60,152)(61,131,71,121)(62,122,72,132)(63,133,73,123)(64,124,74,134)(65,135,75,125)(66,126,76,136)(67,137,77,127)(68,128,78,138)(69,139,79,129)(70,130,80,140)(81,116,91,106)(82,107,92,117)(83,118,93,108)(84,109,94,119)(85,120,95,110)(86,111,96,101)(87,102,97,112)(88,113,98,103)(89,104,99,114)(90,115,100,105), (1,79,86,49)(2,60,87,70)(3,61,88,51)(4,42,89,72)(5,63,90,53)(6,44,91,74)(7,65,92,55)(8,46,93,76)(9,67,94,57)(10,48,95,78)(11,69,96,59)(12,50,97,80)(13,71,98,41)(14,52,99,62)(15,73,100,43)(16,54,81,64)(17,75,82,45)(18,56,83,66)(19,77,84,47)(20,58,85,68)(21,129,111,151)(22,142,112,140)(23,131,113,153)(24,144,114,122)(25,133,115,155)(26,146,116,124)(27,135,117,157)(28,148,118,126)(29,137,119,159)(30,150,120,128)(31,139,101,141)(32,152,102,130)(33,121,103,143)(34,154,104,132)(35,123,105,145)(36,156,106,134)(37,125,107,147)(38,158,108,136)(39,127,109,149)(40,160,110,138), (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)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,76,71,66)(62,65,72,75)(63,74,73,64)(67,70,77,80)(68,79,78,69)(81,90,91,100)(82,99,92,89)(83,88,93,98)(84,97,94,87)(85,86,95,96)(101,110,111,120)(102,119,112,109)(103,108,113,118)(104,117,114,107)(105,106,115,116)(121,136,131,126)(122,125,132,135)(123,134,133,124)(127,130,137,140)(128,139,138,129)(141,160,151,150)(142,149,152,159)(143,158,153,148)(144,147,154,157)(145,156,155,146)>;
G:=Group( (1,21,11,31)(2,32,12,22)(3,23,13,33)(4,34,14,24)(5,25,15,35)(6,36,16,26)(7,27,17,37)(8,38,18,28)(9,29,19,39)(10,40,20,30)(41,143,51,153)(42,154,52,144)(43,145,53,155)(44,156,54,146)(45,147,55,157)(46,158,56,148)(47,149,57,159)(48,160,58,150)(49,151,59,141)(50,142,60,152)(61,131,71,121)(62,122,72,132)(63,133,73,123)(64,124,74,134)(65,135,75,125)(66,126,76,136)(67,137,77,127)(68,128,78,138)(69,139,79,129)(70,130,80,140)(81,116,91,106)(82,107,92,117)(83,118,93,108)(84,109,94,119)(85,120,95,110)(86,111,96,101)(87,102,97,112)(88,113,98,103)(89,104,99,114)(90,115,100,105), (1,79,86,49)(2,60,87,70)(3,61,88,51)(4,42,89,72)(5,63,90,53)(6,44,91,74)(7,65,92,55)(8,46,93,76)(9,67,94,57)(10,48,95,78)(11,69,96,59)(12,50,97,80)(13,71,98,41)(14,52,99,62)(15,73,100,43)(16,54,81,64)(17,75,82,45)(18,56,83,66)(19,77,84,47)(20,58,85,68)(21,129,111,151)(22,142,112,140)(23,131,113,153)(24,144,114,122)(25,133,115,155)(26,146,116,124)(27,135,117,157)(28,148,118,126)(29,137,119,159)(30,150,120,128)(31,139,101,141)(32,152,102,130)(33,121,103,143)(34,154,104,132)(35,123,105,145)(36,156,106,134)(37,125,107,147)(38,158,108,136)(39,127,109,149)(40,160,110,138), (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)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,76,71,66)(62,65,72,75)(63,74,73,64)(67,70,77,80)(68,79,78,69)(81,90,91,100)(82,99,92,89)(83,88,93,98)(84,97,94,87)(85,86,95,96)(101,110,111,120)(102,119,112,109)(103,108,113,118)(104,117,114,107)(105,106,115,116)(121,136,131,126)(122,125,132,135)(123,134,133,124)(127,130,137,140)(128,139,138,129)(141,160,151,150)(142,149,152,159)(143,158,153,148)(144,147,154,157)(145,156,155,146) );
G=PermutationGroup([(1,21,11,31),(2,32,12,22),(3,23,13,33),(4,34,14,24),(5,25,15,35),(6,36,16,26),(7,27,17,37),(8,38,18,28),(9,29,19,39),(10,40,20,30),(41,143,51,153),(42,154,52,144),(43,145,53,155),(44,156,54,146),(45,147,55,157),(46,158,56,148),(47,149,57,159),(48,160,58,150),(49,151,59,141),(50,142,60,152),(61,131,71,121),(62,122,72,132),(63,133,73,123),(64,124,74,134),(65,135,75,125),(66,126,76,136),(67,137,77,127),(68,128,78,138),(69,139,79,129),(70,130,80,140),(81,116,91,106),(82,107,92,117),(83,118,93,108),(84,109,94,119),(85,120,95,110),(86,111,96,101),(87,102,97,112),(88,113,98,103),(89,104,99,114),(90,115,100,105)], [(1,79,86,49),(2,60,87,70),(3,61,88,51),(4,42,89,72),(5,63,90,53),(6,44,91,74),(7,65,92,55),(8,46,93,76),(9,67,94,57),(10,48,95,78),(11,69,96,59),(12,50,97,80),(13,71,98,41),(14,52,99,62),(15,73,100,43),(16,54,81,64),(17,75,82,45),(18,56,83,66),(19,77,84,47),(20,58,85,68),(21,129,111,151),(22,142,112,140),(23,131,113,153),(24,144,114,122),(25,133,115,155),(26,146,116,124),(27,135,117,157),(28,148,118,126),(29,137,119,159),(30,150,120,128),(31,139,101,141),(32,152,102,130),(33,121,103,143),(34,154,104,132),(35,123,105,145),(36,156,106,134),(37,125,107,147),(38,158,108,136),(39,127,109,149),(40,160,110,138)], [(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),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(61,76,71,66),(62,65,72,75),(63,74,73,64),(67,70,77,80),(68,79,78,69),(81,90,91,100),(82,99,92,89),(83,88,93,98),(84,97,94,87),(85,86,95,96),(101,110,111,120),(102,119,112,109),(103,108,113,118),(104,117,114,107),(105,106,115,116),(121,136,131,126),(122,125,132,135),(123,134,133,124),(127,130,137,140),(128,139,138,129),(141,160,151,150),(142,149,152,159),(143,158,153,148),(144,147,154,157),(145,156,155,146)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 34 | 0 |
| 0 | 0 | 0 | 14 | 0 | 34 |
| 0 | 0 | 34 | 0 | 27 | 0 |
| 0 | 0 | 0 | 34 | 0 | 27 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 32 | 17 | 0 | 0 | 0 | 0 |
| 17 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 9 | 24 | 0 | 0 | 0 | 0 |
| 24 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 40 | 34 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,34,0,0,0,0,14,0,34,0,0,34,0,27,0,0,0,0,34,0,27],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[32,17,0,0,0,0,17,9,0,0,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,7,34,0,0,0,0,7,40,0,0],[9,24,0,0,0,0,24,32,0,0,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,7,40,0,0,0,0,7,34,0,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 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 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | D4.10D10 |
| kernel | C42.141D10 | C20⋊2Q8 | C42⋊D5 | Dic5.14D4 | D10.12D4 | C20.17D4 | Dic5⋊Q8 | C5×C4.4D4 | C2×D4⋊2D5 | C2×Q8×D5 | C4×D5 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{141}D_{10} % in TeX
G:=Group("C4^2.141D10"); // GroupNames label
G:=SmallGroup(320,1347);
// by ID
G=gap.SmallGroup(320,1347);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,192,12550]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀