metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.139D10, C10.882- (1+4), (Q8×Dic5)⋊18C2, C4.4D4.8D5, (C4×Dic10)⋊44C2, (C2×D4).169D10, (C2×C20).77C23, (C2×Q8).135D10, C22⋊C4.33D10, (D4×Dic5).15C2, Dic5⋊Q8⋊22C2, C20.124(C4○D4), C4.15(D4⋊2D5), (C4×C20).184C22, (C2×C10).215C24, C23.37(C22×D5), Dic5.44(C4○D4), C20.17D4.11C2, (D4×C10).151C22, C23.D10⋊37C2, C4⋊Dic5.233C22, (C22×C10).45C23, (Q8×C10).124C22, C22.236(C23×D5), Dic5.14D4⋊38C2, C23.D5.52C22, C23.11D10⋊18C2, C5⋊6(C22.50C24), (C4×Dic5).139C22, (C2×Dic5).262C23, C10.D4.48C22, C2.49(D4.10D10), (C2×Dic10).304C22, (C22×Dic5).140C22, C2.74(D5×C4○D4), C10.93(C2×C4○D4), C2.55(C2×D4⋊2D5), (C5×C4.4D4).6C2, (C2×C4).299(C22×D5), (C5×C22⋊C4).62C22, SmallGroup(320,1343)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 614 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], Q8 [×6], C23 [×2], C10 [×3], C10 [×2], C42, C42 [×6], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4, C4.4D4, C42⋊2C2 [×4], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×C10 [×2], C22.50C24, C4×Dic5 [×2], C4×Dic5 [×4], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C22×Dic5 [×2], D4×C10, Q8×C10, C4×Dic10 [×2], C23.11D10 [×2], Dic5.14D4 [×2], C23.D10 [×4], D4×Dic5, C20.17D4, Dic5⋊Q8, Q8×Dic5, C5×C4.4D4, C42.139D10
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], C22.50C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D5×C4○D4, D4.10D10, C42.139D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 125 6 130)(2 121 7 126)(3 127 8 122)(4 123 9 128)(5 129 10 124)(11 140 36 135)(12 136 37 131)(13 132 38 137)(14 138 39 133)(15 134 40 139)(16 115 22 120)(17 111 23 116)(18 117 24 112)(19 113 25 118)(20 119 21 114)(26 102 32 107)(27 108 33 103)(28 104 34 109)(29 110 35 105)(30 106 31 101)(41 100 143 158)(42 159 144 91)(43 92 145 160)(44 151 146 93)(45 94 147 152)(46 153 148 95)(47 96 149 154)(48 155 150 97)(49 98 141 156)(50 157 142 99)(51 70 90 71)(52 72 81 61)(53 62 82 73)(54 74 83 63)(55 64 84 75)(56 76 85 65)(57 66 86 77)(58 78 87 67)(59 68 88 79)(60 80 89 69)
(1 41 36 148)(2 47 37 144)(3 43 38 150)(4 49 39 146)(5 45 40 142)(6 143 11 46)(7 149 12 42)(8 145 13 48)(9 141 14 44)(10 147 15 50)(16 57 27 81)(17 53 28 87)(18 59 29 83)(19 55 30 89)(20 51 26 85)(21 90 32 56)(22 86 33 52)(23 82 34 58)(24 88 35 54)(25 84 31 60)(61 115 66 108)(62 104 67 111)(63 117 68 110)(64 106 69 113)(65 119 70 102)(71 107 76 114)(72 120 77 103)(73 109 78 116)(74 112 79 105)(75 101 80 118)(91 121 96 131)(92 137 97 127)(93 123 98 133)(94 139 99 129)(95 125 100 135)(122 160 132 155)(124 152 134 157)(126 154 136 159)(128 156 138 151)(130 158 140 153)
(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 29 11 24)(2 28 12 23)(3 27 13 22)(4 26 14 21)(5 30 15 25)(6 35 36 18)(7 34 37 17)(8 33 38 16)(9 32 39 20)(10 31 40 19)(41 83 46 88)(42 82 47 87)(43 81 48 86)(44 90 49 85)(45 89 50 84)(51 141 56 146)(52 150 57 145)(53 149 58 144)(54 148 59 143)(55 147 60 142)(61 97 77 160)(62 96 78 159)(63 95 79 158)(64 94 80 157)(65 93 71 156)(66 92 72 155)(67 91 73 154)(68 100 74 153)(69 99 75 152)(70 98 76 151)(101 134 113 129)(102 133 114 128)(103 132 115 127)(104 131 116 126)(105 140 117 125)(106 139 118 124)(107 138 119 123)(108 137 120 122)(109 136 111 121)(110 135 112 130)
G:=sub<Sym(160)| (1,125,6,130)(2,121,7,126)(3,127,8,122)(4,123,9,128)(5,129,10,124)(11,140,36,135)(12,136,37,131)(13,132,38,137)(14,138,39,133)(15,134,40,139)(16,115,22,120)(17,111,23,116)(18,117,24,112)(19,113,25,118)(20,119,21,114)(26,102,32,107)(27,108,33,103)(28,104,34,109)(29,110,35,105)(30,106,31,101)(41,100,143,158)(42,159,144,91)(43,92,145,160)(44,151,146,93)(45,94,147,152)(46,153,148,95)(47,96,149,154)(48,155,150,97)(49,98,141,156)(50,157,142,99)(51,70,90,71)(52,72,81,61)(53,62,82,73)(54,74,83,63)(55,64,84,75)(56,76,85,65)(57,66,86,77)(58,78,87,67)(59,68,88,79)(60,80,89,69), (1,41,36,148)(2,47,37,144)(3,43,38,150)(4,49,39,146)(5,45,40,142)(6,143,11,46)(7,149,12,42)(8,145,13,48)(9,141,14,44)(10,147,15,50)(16,57,27,81)(17,53,28,87)(18,59,29,83)(19,55,30,89)(20,51,26,85)(21,90,32,56)(22,86,33,52)(23,82,34,58)(24,88,35,54)(25,84,31,60)(61,115,66,108)(62,104,67,111)(63,117,68,110)(64,106,69,113)(65,119,70,102)(71,107,76,114)(72,120,77,103)(73,109,78,116)(74,112,79,105)(75,101,80,118)(91,121,96,131)(92,137,97,127)(93,123,98,133)(94,139,99,129)(95,125,100,135)(122,160,132,155)(124,152,134,157)(126,154,136,159)(128,156,138,151)(130,158,140,153), (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), (1,29,11,24)(2,28,12,23)(3,27,13,22)(4,26,14,21)(5,30,15,25)(6,35,36,18)(7,34,37,17)(8,33,38,16)(9,32,39,20)(10,31,40,19)(41,83,46,88)(42,82,47,87)(43,81,48,86)(44,90,49,85)(45,89,50,84)(51,141,56,146)(52,150,57,145)(53,149,58,144)(54,148,59,143)(55,147,60,142)(61,97,77,160)(62,96,78,159)(63,95,79,158)(64,94,80,157)(65,93,71,156)(66,92,72,155)(67,91,73,154)(68,100,74,153)(69,99,75,152)(70,98,76,151)(101,134,113,129)(102,133,114,128)(103,132,115,127)(104,131,116,126)(105,140,117,125)(106,139,118,124)(107,138,119,123)(108,137,120,122)(109,136,111,121)(110,135,112,130)>;
G:=Group( (1,125,6,130)(2,121,7,126)(3,127,8,122)(4,123,9,128)(5,129,10,124)(11,140,36,135)(12,136,37,131)(13,132,38,137)(14,138,39,133)(15,134,40,139)(16,115,22,120)(17,111,23,116)(18,117,24,112)(19,113,25,118)(20,119,21,114)(26,102,32,107)(27,108,33,103)(28,104,34,109)(29,110,35,105)(30,106,31,101)(41,100,143,158)(42,159,144,91)(43,92,145,160)(44,151,146,93)(45,94,147,152)(46,153,148,95)(47,96,149,154)(48,155,150,97)(49,98,141,156)(50,157,142,99)(51,70,90,71)(52,72,81,61)(53,62,82,73)(54,74,83,63)(55,64,84,75)(56,76,85,65)(57,66,86,77)(58,78,87,67)(59,68,88,79)(60,80,89,69), (1,41,36,148)(2,47,37,144)(3,43,38,150)(4,49,39,146)(5,45,40,142)(6,143,11,46)(7,149,12,42)(8,145,13,48)(9,141,14,44)(10,147,15,50)(16,57,27,81)(17,53,28,87)(18,59,29,83)(19,55,30,89)(20,51,26,85)(21,90,32,56)(22,86,33,52)(23,82,34,58)(24,88,35,54)(25,84,31,60)(61,115,66,108)(62,104,67,111)(63,117,68,110)(64,106,69,113)(65,119,70,102)(71,107,76,114)(72,120,77,103)(73,109,78,116)(74,112,79,105)(75,101,80,118)(91,121,96,131)(92,137,97,127)(93,123,98,133)(94,139,99,129)(95,125,100,135)(122,160,132,155)(124,152,134,157)(126,154,136,159)(128,156,138,151)(130,158,140,153), (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), (1,29,11,24)(2,28,12,23)(3,27,13,22)(4,26,14,21)(5,30,15,25)(6,35,36,18)(7,34,37,17)(8,33,38,16)(9,32,39,20)(10,31,40,19)(41,83,46,88)(42,82,47,87)(43,81,48,86)(44,90,49,85)(45,89,50,84)(51,141,56,146)(52,150,57,145)(53,149,58,144)(54,148,59,143)(55,147,60,142)(61,97,77,160)(62,96,78,159)(63,95,79,158)(64,94,80,157)(65,93,71,156)(66,92,72,155)(67,91,73,154)(68,100,74,153)(69,99,75,152)(70,98,76,151)(101,134,113,129)(102,133,114,128)(103,132,115,127)(104,131,116,126)(105,140,117,125)(106,139,118,124)(107,138,119,123)(108,137,120,122)(109,136,111,121)(110,135,112,130) );
G=PermutationGroup([(1,125,6,130),(2,121,7,126),(3,127,8,122),(4,123,9,128),(5,129,10,124),(11,140,36,135),(12,136,37,131),(13,132,38,137),(14,138,39,133),(15,134,40,139),(16,115,22,120),(17,111,23,116),(18,117,24,112),(19,113,25,118),(20,119,21,114),(26,102,32,107),(27,108,33,103),(28,104,34,109),(29,110,35,105),(30,106,31,101),(41,100,143,158),(42,159,144,91),(43,92,145,160),(44,151,146,93),(45,94,147,152),(46,153,148,95),(47,96,149,154),(48,155,150,97),(49,98,141,156),(50,157,142,99),(51,70,90,71),(52,72,81,61),(53,62,82,73),(54,74,83,63),(55,64,84,75),(56,76,85,65),(57,66,86,77),(58,78,87,67),(59,68,88,79),(60,80,89,69)], [(1,41,36,148),(2,47,37,144),(3,43,38,150),(4,49,39,146),(5,45,40,142),(6,143,11,46),(7,149,12,42),(8,145,13,48),(9,141,14,44),(10,147,15,50),(16,57,27,81),(17,53,28,87),(18,59,29,83),(19,55,30,89),(20,51,26,85),(21,90,32,56),(22,86,33,52),(23,82,34,58),(24,88,35,54),(25,84,31,60),(61,115,66,108),(62,104,67,111),(63,117,68,110),(64,106,69,113),(65,119,70,102),(71,107,76,114),(72,120,77,103),(73,109,78,116),(74,112,79,105),(75,101,80,118),(91,121,96,131),(92,137,97,127),(93,123,98,133),(94,139,99,129),(95,125,100,135),(122,160,132,155),(124,152,134,157),(126,154,136,159),(128,156,138,151),(130,158,140,153)], [(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)], [(1,29,11,24),(2,28,12,23),(3,27,13,22),(4,26,14,21),(5,30,15,25),(6,35,36,18),(7,34,37,17),(8,33,38,16),(9,32,39,20),(10,31,40,19),(41,83,46,88),(42,82,47,87),(43,81,48,86),(44,90,49,85),(45,89,50,84),(51,141,56,146),(52,150,57,145),(53,149,58,144),(54,148,59,143),(55,147,60,142),(61,97,77,160),(62,96,78,159),(63,95,79,158),(64,94,80,157),(65,93,71,156),(66,92,72,155),(67,91,73,154),(68,100,74,153),(69,99,75,152),(70,98,76,151),(101,134,113,129),(102,133,114,128),(103,132,115,127),(104,131,116,126),(105,140,117,125),(106,139,118,124),(107,138,119,123),(108,137,120,122),(109,136,111,121),(110,135,112,130)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 9 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 39 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 30 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [40,18,0,0,0,0,9,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,9,32],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,39,0,0,0,0,0,40],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,14,30,0,0,0,0,14,27,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2- (1+4) | D4⋊2D5 | D5×C4○D4 | D4.10D10 |
| kernel | C42.139D10 | C4×Dic10 | C23.11D10 | Dic5.14D4 | C23.D10 | D4×Dic5 | C20.17D4 | Dic5⋊Q8 | Q8×Dic5 | C5×C4.4D4 | C4.4D4 | Dic5 | C20 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{139}D_{10} % in TeX
G:=Group("C4^2.139D10"); // GroupNames label
G:=SmallGroup(320,1343);
// by ID
G=gap.SmallGroup(320,1343);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,387,100,794,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀