metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.157D10, C10.312- (1+4), C10.1352+ (1+4), C20⋊Q8⋊38C2, C4⋊C4.114D10, C42.C2⋊13D5, D10⋊Q8⋊37C2, D10⋊2Q8⋊39C2, C20.6Q8⋊30C2, (C2×C10).243C24, (C2×C20).190C23, (C4×C20).224C22, C4.D20.14C2, (C2×D20).38C22, C2.60(D4⋊8D10), D10.13D4.4C2, C4⋊Dic5.245C22, C22.264(C23×D5), D10⋊C4.43C22, C5⋊4(C22.57C24), (C2×Dic5).125C23, (C4×Dic5).156C22, (C2×Dic10).43C22, C10.D4.86C22, (C22×D5).108C23, C2.61(D4.10D10), C2.32(Q8.10D10), C4⋊C4⋊D5⋊38C2, (C5×C42.C2)⋊16C2, (C2×C4×D5).142C22, (C5×C4⋊C4).198C22, (C2×C4).207(C22×D5), SmallGroup(320,1371)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 710 in 196 conjugacy classes, 91 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic5 [×6], C20 [×7], D10 [×6], C2×C10, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2, C42.C2, C42⋊2C2 [×4], C4⋊Q8 [×2], Dic10 [×3], C4×D5 [×2], D20, C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C20.6Q8, C4.D20, C20⋊Q8 [×2], D10.13D4 [×2], D10⋊Q8 [×2], D10⋊2Q8 [×2], C4⋊C4⋊D5 [×4], C5×C42.C2, C42.157D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D5 [×7], C22.57C24, C23×D5, Q8.10D10, D4⋊8D10, D4.10D10, C42.157D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 149 75 133)(2 160 76 124)(3 151 77 135)(4 142 78 126)(5 153 79 137)(6 144 80 128)(7 155 61 139)(8 146 62 130)(9 157 63 121)(10 148 64 132)(11 159 65 123)(12 150 66 134)(13 141 67 125)(14 152 68 136)(15 143 69 127)(16 154 70 138)(17 145 71 129)(18 156 72 140)(19 147 73 131)(20 158 74 122)(21 49 92 120)(22 60 93 111)(23 51 94 102)(24 42 95 113)(25 53 96 104)(26 44 97 115)(27 55 98 106)(28 46 99 117)(29 57 100 108)(30 48 81 119)(31 59 82 110)(32 50 83 101)(33 41 84 112)(34 52 85 103)(35 43 86 114)(36 54 87 105)(37 45 88 116)(38 56 89 107)(39 47 90 118)(40 58 91 109)
(1 43 65 104)(2 105 66 44)(3 45 67 106)(4 107 68 46)(5 47 69 108)(6 109 70 48)(7 49 71 110)(8 111 72 50)(9 51 73 112)(10 113 74 52)(11 53 75 114)(12 115 76 54)(13 55 77 116)(14 117 78 56)(15 57 79 118)(16 119 80 58)(17 59 61 120)(18 101 62 60)(19 41 63 102)(20 103 64 42)(21 145 82 139)(22 140 83 146)(23 147 84 121)(24 122 85 148)(25 149 86 123)(26 124 87 150)(27 151 88 125)(28 126 89 152)(29 153 90 127)(30 128 91 154)(31 155 92 129)(32 130 93 156)(33 157 94 131)(34 132 95 158)(35 159 96 133)(36 134 97 160)(37 141 98 135)(38 136 99 142)(39 143 100 137)(40 138 81 144)
(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 28 31 38)(22 37 32 27)(23 26 33 36)(24 35 34 25)(29 40 39 30)(41 105 51 115)(42 114 52 104)(43 103 53 113)(44 112 54 102)(45 101 55 111)(46 110 56 120)(47 119 57 109)(48 108 58 118)(49 117 59 107)(50 106 60 116)(61 78 71 68)(62 67 72 77)(63 76 73 66)(64 65 74 75)(69 70 79 80)(81 100 91 90)(82 89 92 99)(83 98 93 88)(84 87 94 97)(85 96 95 86)(121 160 131 150)(122 149 132 159)(123 158 133 148)(124 147 134 157)(125 156 135 146)(126 145 136 155)(127 154 137 144)(128 143 138 153)(129 152 139 142)(130 141 140 151)
G:=sub<Sym(160)| (1,149,75,133)(2,160,76,124)(3,151,77,135)(4,142,78,126)(5,153,79,137)(6,144,80,128)(7,155,61,139)(8,146,62,130)(9,157,63,121)(10,148,64,132)(11,159,65,123)(12,150,66,134)(13,141,67,125)(14,152,68,136)(15,143,69,127)(16,154,70,138)(17,145,71,129)(18,156,72,140)(19,147,73,131)(20,158,74,122)(21,49,92,120)(22,60,93,111)(23,51,94,102)(24,42,95,113)(25,53,96,104)(26,44,97,115)(27,55,98,106)(28,46,99,117)(29,57,100,108)(30,48,81,119)(31,59,82,110)(32,50,83,101)(33,41,84,112)(34,52,85,103)(35,43,86,114)(36,54,87,105)(37,45,88,116)(38,56,89,107)(39,47,90,118)(40,58,91,109), (1,43,65,104)(2,105,66,44)(3,45,67,106)(4,107,68,46)(5,47,69,108)(6,109,70,48)(7,49,71,110)(8,111,72,50)(9,51,73,112)(10,113,74,52)(11,53,75,114)(12,115,76,54)(13,55,77,116)(14,117,78,56)(15,57,79,118)(16,119,80,58)(17,59,61,120)(18,101,62,60)(19,41,63,102)(20,103,64,42)(21,145,82,139)(22,140,83,146)(23,147,84,121)(24,122,85,148)(25,149,86,123)(26,124,87,150)(27,151,88,125)(28,126,89,152)(29,153,90,127)(30,128,91,154)(31,155,92,129)(32,130,93,156)(33,157,94,131)(34,132,95,158)(35,159,96,133)(36,134,97,160)(37,141,98,135)(38,136,99,142)(39,143,100,137)(40,138,81,144), (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)(21,28,31,38)(22,37,32,27)(23,26,33,36)(24,35,34,25)(29,40,39,30)(41,105,51,115)(42,114,52,104)(43,103,53,113)(44,112,54,102)(45,101,55,111)(46,110,56,120)(47,119,57,109)(48,108,58,118)(49,117,59,107)(50,106,60,116)(61,78,71,68)(62,67,72,77)(63,76,73,66)(64,65,74,75)(69,70,79,80)(81,100,91,90)(82,89,92,99)(83,98,93,88)(84,87,94,97)(85,96,95,86)(121,160,131,150)(122,149,132,159)(123,158,133,148)(124,147,134,157)(125,156,135,146)(126,145,136,155)(127,154,137,144)(128,143,138,153)(129,152,139,142)(130,141,140,151)>;
G:=Group( (1,149,75,133)(2,160,76,124)(3,151,77,135)(4,142,78,126)(5,153,79,137)(6,144,80,128)(7,155,61,139)(8,146,62,130)(9,157,63,121)(10,148,64,132)(11,159,65,123)(12,150,66,134)(13,141,67,125)(14,152,68,136)(15,143,69,127)(16,154,70,138)(17,145,71,129)(18,156,72,140)(19,147,73,131)(20,158,74,122)(21,49,92,120)(22,60,93,111)(23,51,94,102)(24,42,95,113)(25,53,96,104)(26,44,97,115)(27,55,98,106)(28,46,99,117)(29,57,100,108)(30,48,81,119)(31,59,82,110)(32,50,83,101)(33,41,84,112)(34,52,85,103)(35,43,86,114)(36,54,87,105)(37,45,88,116)(38,56,89,107)(39,47,90,118)(40,58,91,109), (1,43,65,104)(2,105,66,44)(3,45,67,106)(4,107,68,46)(5,47,69,108)(6,109,70,48)(7,49,71,110)(8,111,72,50)(9,51,73,112)(10,113,74,52)(11,53,75,114)(12,115,76,54)(13,55,77,116)(14,117,78,56)(15,57,79,118)(16,119,80,58)(17,59,61,120)(18,101,62,60)(19,41,63,102)(20,103,64,42)(21,145,82,139)(22,140,83,146)(23,147,84,121)(24,122,85,148)(25,149,86,123)(26,124,87,150)(27,151,88,125)(28,126,89,152)(29,153,90,127)(30,128,91,154)(31,155,92,129)(32,130,93,156)(33,157,94,131)(34,132,95,158)(35,159,96,133)(36,134,97,160)(37,141,98,135)(38,136,99,142)(39,143,100,137)(40,138,81,144), (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)(21,28,31,38)(22,37,32,27)(23,26,33,36)(24,35,34,25)(29,40,39,30)(41,105,51,115)(42,114,52,104)(43,103,53,113)(44,112,54,102)(45,101,55,111)(46,110,56,120)(47,119,57,109)(48,108,58,118)(49,117,59,107)(50,106,60,116)(61,78,71,68)(62,67,72,77)(63,76,73,66)(64,65,74,75)(69,70,79,80)(81,100,91,90)(82,89,92,99)(83,98,93,88)(84,87,94,97)(85,96,95,86)(121,160,131,150)(122,149,132,159)(123,158,133,148)(124,147,134,157)(125,156,135,146)(126,145,136,155)(127,154,137,144)(128,143,138,153)(129,152,139,142)(130,141,140,151) );
G=PermutationGroup([(1,149,75,133),(2,160,76,124),(3,151,77,135),(4,142,78,126),(5,153,79,137),(6,144,80,128),(7,155,61,139),(8,146,62,130),(9,157,63,121),(10,148,64,132),(11,159,65,123),(12,150,66,134),(13,141,67,125),(14,152,68,136),(15,143,69,127),(16,154,70,138),(17,145,71,129),(18,156,72,140),(19,147,73,131),(20,158,74,122),(21,49,92,120),(22,60,93,111),(23,51,94,102),(24,42,95,113),(25,53,96,104),(26,44,97,115),(27,55,98,106),(28,46,99,117),(29,57,100,108),(30,48,81,119),(31,59,82,110),(32,50,83,101),(33,41,84,112),(34,52,85,103),(35,43,86,114),(36,54,87,105),(37,45,88,116),(38,56,89,107),(39,47,90,118),(40,58,91,109)], [(1,43,65,104),(2,105,66,44),(3,45,67,106),(4,107,68,46),(5,47,69,108),(6,109,70,48),(7,49,71,110),(8,111,72,50),(9,51,73,112),(10,113,74,52),(11,53,75,114),(12,115,76,54),(13,55,77,116),(14,117,78,56),(15,57,79,118),(16,119,80,58),(17,59,61,120),(18,101,62,60),(19,41,63,102),(20,103,64,42),(21,145,82,139),(22,140,83,146),(23,147,84,121),(24,122,85,148),(25,149,86,123),(26,124,87,150),(27,151,88,125),(28,126,89,152),(29,153,90,127),(30,128,91,154),(31,155,92,129),(32,130,93,156),(33,157,94,131),(34,132,95,158),(35,159,96,133),(36,134,97,160),(37,141,98,135),(38,136,99,142),(39,143,100,137),(40,138,81,144)], [(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),(21,28,31,38),(22,37,32,27),(23,26,33,36),(24,35,34,25),(29,40,39,30),(41,105,51,115),(42,114,52,104),(43,103,53,113),(44,112,54,102),(45,101,55,111),(46,110,56,120),(47,119,57,109),(48,108,58,118),(49,117,59,107),(50,106,60,116),(61,78,71,68),(62,67,72,77),(63,76,73,66),(64,65,74,75),(69,70,79,80),(81,100,91,90),(82,89,92,99),(83,98,93,88),(84,87,94,97),(85,96,95,86),(121,160,131,150),(122,149,132,159),(123,158,133,148),(124,147,134,157),(125,156,135,146),(126,145,136,155),(127,154,137,144),(128,143,138,153),(129,152,139,142),(130,141,140,151)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 39 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 8 | 32 | 0 |
| 0 | 0 | 0 | 0 | 31 | 5 | 0 | 32 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 20 | 17 | 40 |
| 0 | 0 | 0 | 0 | 39 | 32 | 1 | 24 |
| 31 | 31 | 6 | 35 | 0 | 0 | 0 | 0 |
| 10 | 12 | 6 | 11 | 0 | 0 | 0 | 0 |
| 6 | 35 | 10 | 10 | 0 | 0 | 0 | 0 |
| 6 | 11 | 31 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 1 | 27 | 27 |
| 0 | 0 | 0 | 0 | 15 | 16 | 14 | 2 |
| 0 | 0 | 0 | 0 | 36 | 35 | 11 | 33 |
| 0 | 0 | 0 | 0 | 37 | 33 | 39 | 22 |
| 31 | 31 | 6 | 35 | 0 | 0 | 0 | 0 |
| 12 | 10 | 30 | 35 | 0 | 0 | 0 | 0 |
| 35 | 6 | 31 | 31 | 0 | 0 | 0 | 0 |
| 11 | 6 | 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 1 | 27 | 27 |
| 0 | 0 | 0 | 0 | 0 | 18 | 2 | 14 |
| 0 | 0 | 0 | 0 | 2 | 4 | 33 | 11 |
| 0 | 0 | 0 | 0 | 29 | 26 | 22 | 39 |
G:=sub<GL(8,GF(41))| [39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,9,0,29,31,0,0,0,0,0,9,8,5,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17,1,9,39,0,0,0,0,40,24,20,32,0,0,0,0,0,0,17,1,0,0,0,0,0,0,40,24],[31,10,6,6,0,0,0,0,31,12,35,11,0,0,0,0,6,6,10,31,0,0,0,0,35,11,10,29,0,0,0,0,0,0,0,0,33,15,36,37,0,0,0,0,1,16,35,33,0,0,0,0,27,14,11,39,0,0,0,0,27,2,33,22],[31,12,35,11,0,0,0,0,31,10,6,6,0,0,0,0,6,30,31,12,0,0,0,0,35,35,31,10,0,0,0,0,0,0,0,0,33,0,2,29,0,0,0,0,1,18,4,26,0,0,0,0,27,2,33,22,0,0,0,0,27,14,11,39] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4G | 4H | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | 2+ (1+4) | 2- (1+4) | Q8.10D10 | D4⋊8D10 | D4.10D10 |
| kernel | C42.157D10 | C20.6Q8 | C4.D20 | C20⋊Q8 | D10.13D4 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C42 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 12 | 1 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{157}D_{10} % in TeX
G:=Group("C4^2.157D10"); // GroupNames label
G:=SmallGroup(320,1371);
// by ID
G=gap.SmallGroup(320,1371);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,1571,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^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^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀