metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.176D10, C10.382- (1+4), C4⋊Q8⋊14D5, C4⋊C4.125D10, (Q8×Dic5)⋊23C2, (C2×Q8).148D10, C20.6Q8⋊25C2, C42⋊D5.9C2, C20.138(C4○D4), C4.42(D4⋊2D5), (C4×C20).216C22, (C2×C10).275C24, (C2×C20).108C23, D10⋊3Q8.14C2, Dic5.Q8⋊42C2, C4⋊Dic5.254C22, (Q8×C10).142C22, C22.296(C23×D5), C5⋊7(C22.35C24), (C4×Dic5).172C22, (C2×Dic5).283C23, C10.D4.63C22, (C22×D5).120C23, D10⋊C4.154C22, C2.39(Q8.10D10), (C5×C4⋊Q8)⋊17C2, C4⋊C4⋊D5.4C2, C10.101(C2×C4○D4), C2.65(C2×D4⋊2D5), (C2×C4×D5).157C22, (C5×C4⋊C4).218C22, (C2×C4).221(C22×D5), SmallGroup(320,1403)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 558 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4, C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, D5, C10, C10 [×2], C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C2×Q8 [×2], Dic5 [×7], C20 [×2], C20 [×6], D10 [×3], C2×C10, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22.35C24, C4×Dic5, C4×Dic5 [×4], C10.D4 [×10], C4⋊Dic5 [×6], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5, Q8×C10 [×2], C20.6Q8, C42⋊D5, Dic5.Q8 [×4], C4⋊C4⋊D5 [×4], Q8×Dic5 [×2], D10⋊3Q8 [×2], C5×C4⋊Q8, C42.176D10
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, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, Q8.10D10 [×2], C42.176D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c9 >
►On 160 points
Generators in S160
(1 81 11 91)(2 92 12 82)(3 83 13 93)(4 94 14 84)(5 85 15 95)(6 96 16 86)(7 87 17 97)(8 98 18 88)(9 89 19 99)(10 100 20 90)(21 150 31 160)(22 141 32 151)(23 152 33 142)(24 143 34 153)(25 154 35 144)(26 145 36 155)(27 156 37 146)(28 147 38 157)(29 158 39 148)(30 149 40 159)(41 125 51 135)(42 136 52 126)(43 127 53 137)(44 138 54 128)(45 129 55 139)(46 140 56 130)(47 131 57 121)(48 122 58 132)(49 133 59 123)(50 124 60 134)(61 105 71 115)(62 116 72 106)(63 107 73 117)(64 118 74 108)(65 109 75 119)(66 120 76 110)(67 111 77 101)(68 102 78 112)(69 113 79 103)(70 104 80 114)
(1 32 47 79)(2 80 48 33)(3 34 49 61)(4 62 50 35)(5 36 51 63)(6 64 52 37)(7 38 53 65)(8 66 54 39)(9 40 55 67)(10 68 56 21)(11 22 57 69)(12 70 58 23)(13 24 59 71)(14 72 60 25)(15 26 41 73)(16 74 42 27)(17 28 43 75)(18 76 44 29)(19 30 45 77)(20 78 46 31)(81 151 131 103)(82 104 132 152)(83 153 133 105)(84 106 134 154)(85 155 135 107)(86 108 136 156)(87 157 137 109)(88 110 138 158)(89 159 139 111)(90 112 140 160)(91 141 121 113)(92 114 122 142)(93 143 123 115)(94 116 124 144)(95 145 125 117)(96 118 126 146)(97 147 127 119)(98 120 128 148)(99 149 129 101)(100 102 130 150)
(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 57 46)(2 45 58 9)(3 8 59 44)(4 43 60 7)(5 6 41 42)(11 20 47 56)(12 55 48 19)(13 18 49 54)(14 53 50 17)(15 16 51 52)(21 79 78 22)(23 77 80 40)(24 39 61 76)(25 75 62 38)(26 37 63 74)(27 73 64 36)(28 35 65 72)(29 71 66 34)(30 33 67 70)(31 69 68 32)(81 90 121 130)(82 129 122 89)(83 88 123 128)(84 127 124 87)(85 86 125 126)(91 100 131 140)(92 139 132 99)(93 98 133 138)(94 137 134 97)(95 96 135 136)(101 104 159 142)(102 141 160 103)(105 120 143 158)(106 157 144 119)(107 118 145 156)(108 155 146 117)(109 116 147 154)(110 153 148 115)(111 114 149 152)(112 151 150 113)
G:=sub<Sym(160)| (1,81,11,91)(2,92,12,82)(3,83,13,93)(4,94,14,84)(5,85,15,95)(6,96,16,86)(7,87,17,97)(8,98,18,88)(9,89,19,99)(10,100,20,90)(21,150,31,160)(22,141,32,151)(23,152,33,142)(24,143,34,153)(25,154,35,144)(26,145,36,155)(27,156,37,146)(28,147,38,157)(29,158,39,148)(30,149,40,159)(41,125,51,135)(42,136,52,126)(43,127,53,137)(44,138,54,128)(45,129,55,139)(46,140,56,130)(47,131,57,121)(48,122,58,132)(49,133,59,123)(50,124,60,134)(61,105,71,115)(62,116,72,106)(63,107,73,117)(64,118,74,108)(65,109,75,119)(66,120,76,110)(67,111,77,101)(68,102,78,112)(69,113,79,103)(70,104,80,114), (1,32,47,79)(2,80,48,33)(3,34,49,61)(4,62,50,35)(5,36,51,63)(6,64,52,37)(7,38,53,65)(8,66,54,39)(9,40,55,67)(10,68,56,21)(11,22,57,69)(12,70,58,23)(13,24,59,71)(14,72,60,25)(15,26,41,73)(16,74,42,27)(17,28,43,75)(18,76,44,29)(19,30,45,77)(20,78,46,31)(81,151,131,103)(82,104,132,152)(83,153,133,105)(84,106,134,154)(85,155,135,107)(86,108,136,156)(87,157,137,109)(88,110,138,158)(89,159,139,111)(90,112,140,160)(91,141,121,113)(92,114,122,142)(93,143,123,115)(94,116,124,144)(95,145,125,117)(96,118,126,146)(97,147,127,119)(98,120,128,148)(99,149,129,101)(100,102,130,150), (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), (1,10,57,46)(2,45,58,9)(3,8,59,44)(4,43,60,7)(5,6,41,42)(11,20,47,56)(12,55,48,19)(13,18,49,54)(14,53,50,17)(15,16,51,52)(21,79,78,22)(23,77,80,40)(24,39,61,76)(25,75,62,38)(26,37,63,74)(27,73,64,36)(28,35,65,72)(29,71,66,34)(30,33,67,70)(31,69,68,32)(81,90,121,130)(82,129,122,89)(83,88,123,128)(84,127,124,87)(85,86,125,126)(91,100,131,140)(92,139,132,99)(93,98,133,138)(94,137,134,97)(95,96,135,136)(101,104,159,142)(102,141,160,103)(105,120,143,158)(106,157,144,119)(107,118,145,156)(108,155,146,117)(109,116,147,154)(110,153,148,115)(111,114,149,152)(112,151,150,113)>;
G:=Group( (1,81,11,91)(2,92,12,82)(3,83,13,93)(4,94,14,84)(5,85,15,95)(6,96,16,86)(7,87,17,97)(8,98,18,88)(9,89,19,99)(10,100,20,90)(21,150,31,160)(22,141,32,151)(23,152,33,142)(24,143,34,153)(25,154,35,144)(26,145,36,155)(27,156,37,146)(28,147,38,157)(29,158,39,148)(30,149,40,159)(41,125,51,135)(42,136,52,126)(43,127,53,137)(44,138,54,128)(45,129,55,139)(46,140,56,130)(47,131,57,121)(48,122,58,132)(49,133,59,123)(50,124,60,134)(61,105,71,115)(62,116,72,106)(63,107,73,117)(64,118,74,108)(65,109,75,119)(66,120,76,110)(67,111,77,101)(68,102,78,112)(69,113,79,103)(70,104,80,114), (1,32,47,79)(2,80,48,33)(3,34,49,61)(4,62,50,35)(5,36,51,63)(6,64,52,37)(7,38,53,65)(8,66,54,39)(9,40,55,67)(10,68,56,21)(11,22,57,69)(12,70,58,23)(13,24,59,71)(14,72,60,25)(15,26,41,73)(16,74,42,27)(17,28,43,75)(18,76,44,29)(19,30,45,77)(20,78,46,31)(81,151,131,103)(82,104,132,152)(83,153,133,105)(84,106,134,154)(85,155,135,107)(86,108,136,156)(87,157,137,109)(88,110,138,158)(89,159,139,111)(90,112,140,160)(91,141,121,113)(92,114,122,142)(93,143,123,115)(94,116,124,144)(95,145,125,117)(96,118,126,146)(97,147,127,119)(98,120,128,148)(99,149,129,101)(100,102,130,150), (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), (1,10,57,46)(2,45,58,9)(3,8,59,44)(4,43,60,7)(5,6,41,42)(11,20,47,56)(12,55,48,19)(13,18,49,54)(14,53,50,17)(15,16,51,52)(21,79,78,22)(23,77,80,40)(24,39,61,76)(25,75,62,38)(26,37,63,74)(27,73,64,36)(28,35,65,72)(29,71,66,34)(30,33,67,70)(31,69,68,32)(81,90,121,130)(82,129,122,89)(83,88,123,128)(84,127,124,87)(85,86,125,126)(91,100,131,140)(92,139,132,99)(93,98,133,138)(94,137,134,97)(95,96,135,136)(101,104,159,142)(102,141,160,103)(105,120,143,158)(106,157,144,119)(107,118,145,156)(108,155,146,117)(109,116,147,154)(110,153,148,115)(111,114,149,152)(112,151,150,113) );
G=PermutationGroup([(1,81,11,91),(2,92,12,82),(3,83,13,93),(4,94,14,84),(5,85,15,95),(6,96,16,86),(7,87,17,97),(8,98,18,88),(9,89,19,99),(10,100,20,90),(21,150,31,160),(22,141,32,151),(23,152,33,142),(24,143,34,153),(25,154,35,144),(26,145,36,155),(27,156,37,146),(28,147,38,157),(29,158,39,148),(30,149,40,159),(41,125,51,135),(42,136,52,126),(43,127,53,137),(44,138,54,128),(45,129,55,139),(46,140,56,130),(47,131,57,121),(48,122,58,132),(49,133,59,123),(50,124,60,134),(61,105,71,115),(62,116,72,106),(63,107,73,117),(64,118,74,108),(65,109,75,119),(66,120,76,110),(67,111,77,101),(68,102,78,112),(69,113,79,103),(70,104,80,114)], [(1,32,47,79),(2,80,48,33),(3,34,49,61),(4,62,50,35),(5,36,51,63),(6,64,52,37),(7,38,53,65),(8,66,54,39),(9,40,55,67),(10,68,56,21),(11,22,57,69),(12,70,58,23),(13,24,59,71),(14,72,60,25),(15,26,41,73),(16,74,42,27),(17,28,43,75),(18,76,44,29),(19,30,45,77),(20,78,46,31),(81,151,131,103),(82,104,132,152),(83,153,133,105),(84,106,134,154),(85,155,135,107),(86,108,136,156),(87,157,137,109),(88,110,138,158),(89,159,139,111),(90,112,140,160),(91,141,121,113),(92,114,122,142),(93,143,123,115),(94,116,124,144),(95,145,125,117),(96,118,126,146),(97,147,127,119),(98,120,128,148),(99,149,129,101),(100,102,130,150)], [(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)], [(1,10,57,46),(2,45,58,9),(3,8,59,44),(4,43,60,7),(5,6,41,42),(11,20,47,56),(12,55,48,19),(13,18,49,54),(14,53,50,17),(15,16,51,52),(21,79,78,22),(23,77,80,40),(24,39,61,76),(25,75,62,38),(26,37,63,74),(27,73,64,36),(28,35,65,72),(29,71,66,34),(30,33,67,70),(31,69,68,32),(81,90,121,130),(82,129,122,89),(83,88,123,128),(84,127,124,87),(85,86,125,126),(91,100,131,140),(92,139,132,99),(93,98,133,138),(94,137,134,97),(95,96,135,136),(101,104,159,142),(102,141,160,103),(105,120,143,158),(106,157,144,119),(107,118,145,156),(108,155,146,117),(109,116,147,154),(110,153,148,115),(111,114,149,152),(112,151,150,113)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 0 | 35 | 0 |
| 0 | 0 | 0 | 0 | 0 | 39 | 0 | 35 |
| 0 | 0 | 0 | 0 | 35 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 | 0 | 2 |
| 17 | 40 | 14 | 11 | 0 | 0 | 0 | 0 |
| 1 | 24 | 14 | 14 | 0 | 0 | 0 | 0 |
| 13 | 22 | 17 | 1 | 0 | 0 | 0 | 0 |
| 28 | 13 | 40 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 24 |
| 10 | 23 | 0 | 11 | 0 | 0 | 0 | 0 |
| 18 | 13 | 30 | 0 | 0 | 0 | 0 | 0 |
| 34 | 35 | 28 | 18 | 0 | 0 | 0 | 0 |
| 39 | 34 | 23 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 7 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 | 0 | 0 |
| 31 | 18 | 0 | 30 | 0 | 0 | 0 | 0 |
| 6 | 10 | 30 | 36 | 0 | 0 | 0 | 0 |
| 10 | 19 | 28 | 32 | 0 | 0 | 0 | 0 |
| 19 | 0 | 23 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 | 0 | 0 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,39,0,35,0,0,0,0,0,0,39,0,35,0,0,0,0,35,0,2,0,0,0,0,0,0,35,0,2],[17,1,13,28,0,0,0,0,40,24,22,13,0,0,0,0,14,14,17,40,0,0,0,0,11,14,1,24,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,35,24,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,35,24],[10,18,34,39,0,0,0,0,23,13,35,34,0,0,0,0,0,30,28,23,0,0,0,0,11,0,18,31,0,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0,0,0,0,0,34,7,0,0,0,0,0,0,35,0,0,0],[31,6,10,19,0,0,0,0,18,10,19,0,0,0,0,0,0,30,28,23,0,0,0,0,30,36,32,13,0,0,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,40,7,0,0,0,0,7,34,0,0,0,0,0,0,1,34,0,0] >;
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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- (1+4) | D4⋊2D5 | Q8.10D10 |
| kernel | C42.176D10 | C20.6Q8 | C42⋊D5 | Dic5.Q8 | C4⋊C4⋊D5 | Q8×Dic5 | D10⋊3Q8 | C5×C4⋊Q8 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 2 | 4 | 2 | 8 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{176}D_{10} % in TeX
G:=Group("C4^2.176D10"); // GroupNames label
G:=SmallGroup(320,1403);
// by ID
G=gap.SmallGroup(320,1403);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀