metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.162D10, C10.1012- (1+4), C4⋊C4.213D10, C42⋊2C2⋊5D5, C42⋊D5⋊7C2, D10⋊Q8⋊44C2, (C4×Dic10)⋊15C2, (C4×C20).34C22, C22⋊C4.80D10, C4.Dic10⋊40C2, D10.41(C4○D4), (C2×C20).604C23, (C2×C10).252C24, Dic5⋊4D4.5C2, C23.58(C22×D5), Dic5.48(C4○D4), Dic5.Q8⋊38C2, D10.12D4.4C2, C23.D10⋊46C2, C4⋊Dic5.247C22, (C22×C10).66C23, C22.273(C23×D5), Dic5.14D4⋊46C2, C23.D5.68C22, C23.11D10⋊22C2, (C4×Dic5).238C22, (C2×Dic5).130C23, (C22×D5).236C23, C2.65(D4.10D10), D10⋊C4.140C22, C5⋊11(C22.46C24), (C2×Dic10).263C22, C10.D4.146C22, (C22×Dic5).152C22, (D5×C4⋊C4)⋊42C2, C2.99(D5×C4○D4), C4⋊C4⋊7D5⋊41C2, (C5×C42⋊2C2)⋊7C2, C10.210(C2×C4○D4), (C2×C4×D5).271C22, (C2×C4).88(C22×D5), (C5×C4⋊C4).204C22, (C2×C5⋊D4).72C22, (C5×C22⋊C4).77C22, SmallGroup(320,1380)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 678 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×7], C5, C2×C4 [×6], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42, C42 [×4], C22⋊C4 [×3], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×6], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2, C42⋊2C2, Dic10 [×2], C4×D5 [×6], C2×Dic5 [×7], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×6], C22×D5, C22×C10, C22.46C24, C4×Dic5 [×4], C10.D4 [×9], C4⋊Dic5 [×4], D10⋊C4 [×3], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×3], C22×Dic5, C2×C5⋊D4, C4×Dic10, C42⋊D5, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊4D4, D10.12D4 [×2], Dic5.Q8 [×2], C4.Dic10, D5×C4⋊C4, C4⋊C4⋊7D5, D10⋊Q8, C5×C42⋊2C2, C42.162D10
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], C22.46C24, C23×D5, D5×C4○D4 [×2], D4.10D10, C42.162D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 139 49 39)(2 130 50 30)(3 121 51 21)(4 132 52 32)(5 123 53 23)(6 134 54 34)(7 125 55 25)(8 136 56 36)(9 127 57 27)(10 138 58 38)(11 129 59 29)(12 140 60 40)(13 131 41 31)(14 122 42 22)(15 133 43 33)(16 124 44 24)(17 135 45 35)(18 126 46 26)(19 137 47 37)(20 128 48 28)(61 100 154 108)(62 91 155 119)(63 82 156 110)(64 93 157 101)(65 84 158 112)(66 95 159 103)(67 86 160 114)(68 97 141 105)(69 88 142 116)(70 99 143 107)(71 90 144 118)(72 81 145 109)(73 92 146 120)(74 83 147 111)(75 94 148 102)(76 85 149 113)(77 96 150 104)(78 87 151 115)(79 98 152 106)(80 89 153 117)
(1 155 11 145)(2 73 12 63)(3 157 13 147)(4 75 14 65)(5 159 15 149)(6 77 16 67)(7 141 17 151)(8 79 18 69)(9 143 19 153)(10 61 20 71)(21 93 31 83)(22 112 32 102)(23 95 33 85)(24 114 34 104)(25 97 35 87)(26 116 36 106)(27 99 37 89)(28 118 38 108)(29 81 39 91)(30 120 40 110)(41 74 51 64)(42 158 52 148)(43 76 53 66)(44 160 54 150)(45 78 55 68)(46 142 56 152)(47 80 57 70)(48 144 58 154)(49 62 59 72)(50 146 60 156)(82 130 92 140)(84 132 94 122)(86 134 96 124)(88 136 98 126)(90 138 100 128)(101 131 111 121)(103 133 113 123)(105 135 115 125)(107 137 117 127)(109 139 119 129)
(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 26 31 36)(22 35 32 25)(23 24 33 34)(27 40 37 30)(28 29 38 39)(41 46 51 56)(42 55 52 45)(43 44 53 54)(47 60 57 50)(48 49 58 59)(61 155 71 145)(62 144 72 154)(63 153 73 143)(64 142 74 152)(65 151 75 141)(66 160 76 150)(67 149 77 159)(68 158 78 148)(69 147 79 157)(70 156 80 146)(81 118 91 108)(82 107 92 117)(83 116 93 106)(84 105 94 115)(85 114 95 104)(86 103 96 113)(87 112 97 102)(88 101 98 111)(89 110 99 120)(90 119 100 109)(121 126 131 136)(122 135 132 125)(123 124 133 134)(127 140 137 130)(128 129 138 139)
G:=sub<Sym(160)| (1,139,49,39)(2,130,50,30)(3,121,51,21)(4,132,52,32)(5,123,53,23)(6,134,54,34)(7,125,55,25)(8,136,56,36)(9,127,57,27)(10,138,58,38)(11,129,59,29)(12,140,60,40)(13,131,41,31)(14,122,42,22)(15,133,43,33)(16,124,44,24)(17,135,45,35)(18,126,46,26)(19,137,47,37)(20,128,48,28)(61,100,154,108)(62,91,155,119)(63,82,156,110)(64,93,157,101)(65,84,158,112)(66,95,159,103)(67,86,160,114)(68,97,141,105)(69,88,142,116)(70,99,143,107)(71,90,144,118)(72,81,145,109)(73,92,146,120)(74,83,147,111)(75,94,148,102)(76,85,149,113)(77,96,150,104)(78,87,151,115)(79,98,152,106)(80,89,153,117), (1,155,11,145)(2,73,12,63)(3,157,13,147)(4,75,14,65)(5,159,15,149)(6,77,16,67)(7,141,17,151)(8,79,18,69)(9,143,19,153)(10,61,20,71)(21,93,31,83)(22,112,32,102)(23,95,33,85)(24,114,34,104)(25,97,35,87)(26,116,36,106)(27,99,37,89)(28,118,38,108)(29,81,39,91)(30,120,40,110)(41,74,51,64)(42,158,52,148)(43,76,53,66)(44,160,54,150)(45,78,55,68)(46,142,56,152)(47,80,57,70)(48,144,58,154)(49,62,59,72)(50,146,60,156)(82,130,92,140)(84,132,94,122)(86,134,96,124)(88,136,98,126)(90,138,100,128)(101,131,111,121)(103,133,113,123)(105,135,115,125)(107,137,117,127)(109,139,119,129), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,46,51,56)(42,55,52,45)(43,44,53,54)(47,60,57,50)(48,49,58,59)(61,155,71,145)(62,144,72,154)(63,153,73,143)(64,142,74,152)(65,151,75,141)(66,160,76,150)(67,149,77,159)(68,158,78,148)(69,147,79,157)(70,156,80,146)(81,118,91,108)(82,107,92,117)(83,116,93,106)(84,105,94,115)(85,114,95,104)(86,103,96,113)(87,112,97,102)(88,101,98,111)(89,110,99,120)(90,119,100,109)(121,126,131,136)(122,135,132,125)(123,124,133,134)(127,140,137,130)(128,129,138,139)>;
G:=Group( (1,139,49,39)(2,130,50,30)(3,121,51,21)(4,132,52,32)(5,123,53,23)(6,134,54,34)(7,125,55,25)(8,136,56,36)(9,127,57,27)(10,138,58,38)(11,129,59,29)(12,140,60,40)(13,131,41,31)(14,122,42,22)(15,133,43,33)(16,124,44,24)(17,135,45,35)(18,126,46,26)(19,137,47,37)(20,128,48,28)(61,100,154,108)(62,91,155,119)(63,82,156,110)(64,93,157,101)(65,84,158,112)(66,95,159,103)(67,86,160,114)(68,97,141,105)(69,88,142,116)(70,99,143,107)(71,90,144,118)(72,81,145,109)(73,92,146,120)(74,83,147,111)(75,94,148,102)(76,85,149,113)(77,96,150,104)(78,87,151,115)(79,98,152,106)(80,89,153,117), (1,155,11,145)(2,73,12,63)(3,157,13,147)(4,75,14,65)(5,159,15,149)(6,77,16,67)(7,141,17,151)(8,79,18,69)(9,143,19,153)(10,61,20,71)(21,93,31,83)(22,112,32,102)(23,95,33,85)(24,114,34,104)(25,97,35,87)(26,116,36,106)(27,99,37,89)(28,118,38,108)(29,81,39,91)(30,120,40,110)(41,74,51,64)(42,158,52,148)(43,76,53,66)(44,160,54,150)(45,78,55,68)(46,142,56,152)(47,80,57,70)(48,144,58,154)(49,62,59,72)(50,146,60,156)(82,130,92,140)(84,132,94,122)(86,134,96,124)(88,136,98,126)(90,138,100,128)(101,131,111,121)(103,133,113,123)(105,135,115,125)(107,137,117,127)(109,139,119,129), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,46,51,56)(42,55,52,45)(43,44,53,54)(47,60,57,50)(48,49,58,59)(61,155,71,145)(62,144,72,154)(63,153,73,143)(64,142,74,152)(65,151,75,141)(66,160,76,150)(67,149,77,159)(68,158,78,148)(69,147,79,157)(70,156,80,146)(81,118,91,108)(82,107,92,117)(83,116,93,106)(84,105,94,115)(85,114,95,104)(86,103,96,113)(87,112,97,102)(88,101,98,111)(89,110,99,120)(90,119,100,109)(121,126,131,136)(122,135,132,125)(123,124,133,134)(127,140,137,130)(128,129,138,139) );
G=PermutationGroup([(1,139,49,39),(2,130,50,30),(3,121,51,21),(4,132,52,32),(5,123,53,23),(6,134,54,34),(7,125,55,25),(8,136,56,36),(9,127,57,27),(10,138,58,38),(11,129,59,29),(12,140,60,40),(13,131,41,31),(14,122,42,22),(15,133,43,33),(16,124,44,24),(17,135,45,35),(18,126,46,26),(19,137,47,37),(20,128,48,28),(61,100,154,108),(62,91,155,119),(63,82,156,110),(64,93,157,101),(65,84,158,112),(66,95,159,103),(67,86,160,114),(68,97,141,105),(69,88,142,116),(70,99,143,107),(71,90,144,118),(72,81,145,109),(73,92,146,120),(74,83,147,111),(75,94,148,102),(76,85,149,113),(77,96,150,104),(78,87,151,115),(79,98,152,106),(80,89,153,117)], [(1,155,11,145),(2,73,12,63),(3,157,13,147),(4,75,14,65),(5,159,15,149),(6,77,16,67),(7,141,17,151),(8,79,18,69),(9,143,19,153),(10,61,20,71),(21,93,31,83),(22,112,32,102),(23,95,33,85),(24,114,34,104),(25,97,35,87),(26,116,36,106),(27,99,37,89),(28,118,38,108),(29,81,39,91),(30,120,40,110),(41,74,51,64),(42,158,52,148),(43,76,53,66),(44,160,54,150),(45,78,55,68),(46,142,56,152),(47,80,57,70),(48,144,58,154),(49,62,59,72),(50,146,60,156),(82,130,92,140),(84,132,94,122),(86,134,96,124),(88,136,98,126),(90,138,100,128),(101,131,111,121),(103,133,113,123),(105,135,115,125),(107,137,117,127),(109,139,119,129)], [(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),(21,26,31,36),(22,35,32,25),(23,24,33,34),(27,40,37,30),(28,29,38,39),(41,46,51,56),(42,55,52,45),(43,44,53,54),(47,60,57,50),(48,49,58,59),(61,155,71,145),(62,144,72,154),(63,153,73,143),(64,142,74,152),(65,151,75,141),(66,160,76,150),(67,149,77,159),(68,158,78,148),(69,147,79,157),(70,156,80,146),(81,118,91,108),(82,107,92,117),(83,116,93,106),(84,105,94,115),(85,114,95,104),(86,103,96,113),(87,112,97,102),(88,101,98,111),(89,110,99,120),(90,119,100,109),(121,126,131,136),(122,135,132,125),(123,124,133,134),(127,140,137,130),(128,129,138,139)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 23 |
| 0 | 0 | 0 | 0 | 9 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 22 |
| 0 | 0 | 0 | 0 | 40 | 29 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 22 |
| 0 | 0 | 0 | 0 | 40 | 29 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,9,0,0,0,0,23,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,12,40,0,0,0,0,22,29],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,40,0,0,0,0,22,29] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 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 | 4 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | 2- (1+4) | D5×C4○D4 | D4.10D10 |
| kernel | C42.162D10 | C4×Dic10 | C42⋊D5 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.Q8 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10⋊Q8 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | D10 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 6 | 6 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{162}D_{10} % in TeX
G:=Group("C4^2.162D10"); // GroupNames label
G:=SmallGroup(320,1380);
// by ID
G=gap.SmallGroup(320,1380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,570,80,12550]);
// 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,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀