metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.148D10, C10.952- (1+4), C10.1302+ (1+4), (C4×D5)⋊1Q8, C20⋊Q8⋊35C2, C4.39(Q8×D5), D10.4(C2×Q8), C20.50(C2×Q8), C42.C2⋊4D5, C4⋊C4.111D10, C20⋊2Q8⋊32C2, Dic5.6(C2×Q8), (C2×C20).87C23, D10⋊Q8.2C2, C4.Dic10⋊33C2, C42⋊D5.6C2, C10.42(C22×Q8), (C2×C10).233C24, (C4×C20).193C22, D10⋊2Q8.12C2, C2.55(D4⋊8D10), Dic5.Q8⋊32C2, C4⋊Dic5.240C22, C22.254(C23×D5), D10⋊C4.39C22, C5⋊4(C23.41C23), (C2×Dic10).42C22, (C4×Dic5).148C22, (C2×Dic5).121C23, (C22×D5).230C23, C2.57(D4.10D10), C10.D4.122C22, C2.25(C2×Q8×D5), (D5×C4⋊C4).11C2, (C5×C42.C2)⋊6C2, C4⋊C4⋊7D5.12C2, (C2×C4×D5).133C22, (C5×C4⋊C4).188C22, (C2×C4).203(C22×D5), SmallGroup(320,1361)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 686 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×14], C22×C4 [×3], C2×Q8 [×4], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2, C42.C2 [×3], C4⋊Q8 [×4], Dic10 [×4], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C22×D5, C23.41C23, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10 [×4], C2×C4×D5, C2×C4×D5 [×2], C20⋊2Q8, C42⋊D5, C20⋊Q8, C20⋊Q8 [×2], Dic5.Q8 [×2], C4.Dic10, D5×C4⋊C4, C4⋊C4⋊7D5, D10⋊Q8 [×2], D10⋊2Q8 [×2], C5×C42.C2, C42.148D10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ (1+4), 2- (1+4), C22×D5 [×7], C23.41C23, Q8×D5 [×2], C23×D5, C2×Q8×D5, D4⋊8D10, D4.10D10, C42.148D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, 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 24 131 61)(2 62 132 25)(3 26 133 63)(4 64 134 27)(5 28 135 65)(6 66 136 29)(7 30 137 67)(8 68 138 31)(9 32 139 69)(10 70 140 33)(11 34 121 71)(12 72 122 35)(13 36 123 73)(14 74 124 37)(15 38 125 75)(16 76 126 39)(17 40 127 77)(18 78 128 21)(19 22 129 79)(20 80 130 23)(41 110 151 100)(42 81 152 111)(43 112 153 82)(44 83 154 113)(45 114 155 84)(46 85 156 115)(47 116 157 86)(48 87 158 117)(49 118 159 88)(50 89 160 119)(51 120 141 90)(52 91 142 101)(53 102 143 92)(54 93 144 103)(55 104 145 94)(56 95 146 105)(57 106 147 96)(58 97 148 107)(59 108 149 98)(60 99 150 109)
(1 59 11 49)(2 160 12 150)(3 41 13 51)(4 142 14 152)(5 43 15 53)(6 144 16 154)(7 45 17 55)(8 146 18 156)(9 47 19 57)(10 148 20 158)(21 85 31 95)(22 106 32 116)(23 87 33 97)(24 108 34 118)(25 89 35 99)(26 110 36 120)(27 91 37 81)(28 112 38 102)(29 93 39 83)(30 114 40 104)(42 134 52 124)(44 136 54 126)(46 138 56 128)(48 140 58 130)(50 122 60 132)(61 98 71 88)(62 119 72 109)(63 100 73 90)(64 101 74 111)(65 82 75 92)(66 103 76 113)(67 84 77 94)(68 105 78 115)(69 86 79 96)(70 107 80 117)(121 159 131 149)(123 141 133 151)(125 143 135 153)(127 145 137 155)(129 147 139 157)
(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)(41 156 51 146)(42 145 52 155)(43 154 53 144)(44 143 54 153)(45 152 55 142)(46 141 56 151)(47 150 57 160)(48 159 58 149)(49 148 59 158)(50 157 60 147)(61 70 71 80)(62 79 72 69)(63 68 73 78)(64 77 74 67)(65 66 75 76)(81 104 91 114)(82 113 92 103)(83 102 93 112)(84 111 94 101)(85 120 95 110)(86 109 96 119)(87 118 97 108)(88 107 98 117)(89 116 99 106)(90 105 100 115)(121 130 131 140)(122 139 132 129)(123 128 133 138)(124 137 134 127)(125 126 135 136)
G:=sub<Sym(160)| (1,24,131,61)(2,62,132,25)(3,26,133,63)(4,64,134,27)(5,28,135,65)(6,66,136,29)(7,30,137,67)(8,68,138,31)(9,32,139,69)(10,70,140,33)(11,34,121,71)(12,72,122,35)(13,36,123,73)(14,74,124,37)(15,38,125,75)(16,76,126,39)(17,40,127,77)(18,78,128,21)(19,22,129,79)(20,80,130,23)(41,110,151,100)(42,81,152,111)(43,112,153,82)(44,83,154,113)(45,114,155,84)(46,85,156,115)(47,116,157,86)(48,87,158,117)(49,118,159,88)(50,89,160,119)(51,120,141,90)(52,91,142,101)(53,102,143,92)(54,93,144,103)(55,104,145,94)(56,95,146,105)(57,106,147,96)(58,97,148,107)(59,108,149,98)(60,99,150,109), (1,59,11,49)(2,160,12,150)(3,41,13,51)(4,142,14,152)(5,43,15,53)(6,144,16,154)(7,45,17,55)(8,146,18,156)(9,47,19,57)(10,148,20,158)(21,85,31,95)(22,106,32,116)(23,87,33,97)(24,108,34,118)(25,89,35,99)(26,110,36,120)(27,91,37,81)(28,112,38,102)(29,93,39,83)(30,114,40,104)(42,134,52,124)(44,136,54,126)(46,138,56,128)(48,140,58,130)(50,122,60,132)(61,98,71,88)(62,119,72,109)(63,100,73,90)(64,101,74,111)(65,82,75,92)(66,103,76,113)(67,84,77,94)(68,105,78,115)(69,86,79,96)(70,107,80,117)(121,159,131,149)(123,141,133,151)(125,143,135,153)(127,145,137,155)(129,147,139,157), (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)(41,156,51,146)(42,145,52,155)(43,154,53,144)(44,143,54,153)(45,152,55,142)(46,141,56,151)(47,150,57,160)(48,159,58,149)(49,148,59,158)(50,157,60,147)(61,70,71,80)(62,79,72,69)(63,68,73,78)(64,77,74,67)(65,66,75,76)(81,104,91,114)(82,113,92,103)(83,102,93,112)(84,111,94,101)(85,120,95,110)(86,109,96,119)(87,118,97,108)(88,107,98,117)(89,116,99,106)(90,105,100,115)(121,130,131,140)(122,139,132,129)(123,128,133,138)(124,137,134,127)(125,126,135,136)>;
G:=Group( (1,24,131,61)(2,62,132,25)(3,26,133,63)(4,64,134,27)(5,28,135,65)(6,66,136,29)(7,30,137,67)(8,68,138,31)(9,32,139,69)(10,70,140,33)(11,34,121,71)(12,72,122,35)(13,36,123,73)(14,74,124,37)(15,38,125,75)(16,76,126,39)(17,40,127,77)(18,78,128,21)(19,22,129,79)(20,80,130,23)(41,110,151,100)(42,81,152,111)(43,112,153,82)(44,83,154,113)(45,114,155,84)(46,85,156,115)(47,116,157,86)(48,87,158,117)(49,118,159,88)(50,89,160,119)(51,120,141,90)(52,91,142,101)(53,102,143,92)(54,93,144,103)(55,104,145,94)(56,95,146,105)(57,106,147,96)(58,97,148,107)(59,108,149,98)(60,99,150,109), (1,59,11,49)(2,160,12,150)(3,41,13,51)(4,142,14,152)(5,43,15,53)(6,144,16,154)(7,45,17,55)(8,146,18,156)(9,47,19,57)(10,148,20,158)(21,85,31,95)(22,106,32,116)(23,87,33,97)(24,108,34,118)(25,89,35,99)(26,110,36,120)(27,91,37,81)(28,112,38,102)(29,93,39,83)(30,114,40,104)(42,134,52,124)(44,136,54,126)(46,138,56,128)(48,140,58,130)(50,122,60,132)(61,98,71,88)(62,119,72,109)(63,100,73,90)(64,101,74,111)(65,82,75,92)(66,103,76,113)(67,84,77,94)(68,105,78,115)(69,86,79,96)(70,107,80,117)(121,159,131,149)(123,141,133,151)(125,143,135,153)(127,145,137,155)(129,147,139,157), (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)(41,156,51,146)(42,145,52,155)(43,154,53,144)(44,143,54,153)(45,152,55,142)(46,141,56,151)(47,150,57,160)(48,159,58,149)(49,148,59,158)(50,157,60,147)(61,70,71,80)(62,79,72,69)(63,68,73,78)(64,77,74,67)(65,66,75,76)(81,104,91,114)(82,113,92,103)(83,102,93,112)(84,111,94,101)(85,120,95,110)(86,109,96,119)(87,118,97,108)(88,107,98,117)(89,116,99,106)(90,105,100,115)(121,130,131,140)(122,139,132,129)(123,128,133,138)(124,137,134,127)(125,126,135,136) );
G=PermutationGroup([(1,24,131,61),(2,62,132,25),(3,26,133,63),(4,64,134,27),(5,28,135,65),(6,66,136,29),(7,30,137,67),(8,68,138,31),(9,32,139,69),(10,70,140,33),(11,34,121,71),(12,72,122,35),(13,36,123,73),(14,74,124,37),(15,38,125,75),(16,76,126,39),(17,40,127,77),(18,78,128,21),(19,22,129,79),(20,80,130,23),(41,110,151,100),(42,81,152,111),(43,112,153,82),(44,83,154,113),(45,114,155,84),(46,85,156,115),(47,116,157,86),(48,87,158,117),(49,118,159,88),(50,89,160,119),(51,120,141,90),(52,91,142,101),(53,102,143,92),(54,93,144,103),(55,104,145,94),(56,95,146,105),(57,106,147,96),(58,97,148,107),(59,108,149,98),(60,99,150,109)], [(1,59,11,49),(2,160,12,150),(3,41,13,51),(4,142,14,152),(5,43,15,53),(6,144,16,154),(7,45,17,55),(8,146,18,156),(9,47,19,57),(10,148,20,158),(21,85,31,95),(22,106,32,116),(23,87,33,97),(24,108,34,118),(25,89,35,99),(26,110,36,120),(27,91,37,81),(28,112,38,102),(29,93,39,83),(30,114,40,104),(42,134,52,124),(44,136,54,126),(46,138,56,128),(48,140,58,130),(50,122,60,132),(61,98,71,88),(62,119,72,109),(63,100,73,90),(64,101,74,111),(65,82,75,92),(66,103,76,113),(67,84,77,94),(68,105,78,115),(69,86,79,96),(70,107,80,117),(121,159,131,149),(123,141,133,151),(125,143,135,153),(127,145,137,155),(129,147,139,157)], [(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),(41,156,51,146),(42,145,52,155),(43,154,53,144),(44,143,54,153),(45,152,55,142),(46,141,56,151),(47,150,57,160),(48,159,58,149),(49,148,59,158),(50,157,60,147),(61,70,71,80),(62,79,72,69),(63,68,73,78),(64,77,74,67),(65,66,75,76),(81,104,91,114),(82,113,92,103),(83,102,93,112),(84,111,94,101),(85,120,95,110),(86,109,96,119),(87,118,97,108),(88,107,98,117),(89,116,99,106),(90,105,100,115),(121,130,131,140),(122,139,132,129),(123,128,133,138),(124,137,134,127),(125,126,135,136)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 7 | 9 | 37 |
| 0 | 0 | 0 | 34 | 32 | 9 |
| 0 | 0 | 26 | 7 | 22 | 0 |
| 0 | 0 | 26 | 6 | 15 | 7 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 28 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 0 | 0 | 33 | 39 | 28 |
| 0 | 0 | 33 | 34 | 13 | 2 |
| 6 | 39 | 0 | 0 | 0 | 0 |
| 39 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 | 17 | 13 |
| 0 | 0 | 1 | 0 | 8 | 17 |
| 0 | 0 | 21 | 10 | 1 | 1 |
| 0 | 0 | 34 | 19 | 13 | 6 |
| 35 | 2 | 0 | 0 | 0 | 0 |
| 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 1 | 13 | 17 |
| 0 | 0 | 1 | 34 | 17 | 8 |
| 0 | 0 | 21 | 10 | 1 | 1 |
| 0 | 0 | 34 | 39 | 6 | 13 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,0,26,26,0,0,7,34,7,6,0,0,9,32,22,15,0,0,37,9,0,7],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,2,13,0,33,0,0,28,39,33,34,0,0,0,0,39,13,0,0,0,0,28,2],[6,39,0,0,0,0,39,35,0,0,0,0,0,0,34,1,21,34,0,0,7,0,10,19,0,0,17,8,1,13,0,0,13,17,1,6],[35,2,0,0,0,0,2,6,0,0,0,0,0,0,34,1,21,34,0,0,1,34,10,39,0,0,13,17,1,6,0,0,17,8,1,13] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | 2+ (1+4) | 2- (1+4) | Q8×D5 | D4⋊8D10 | D4.10D10 |
| kernel | C42.148D10 | C20⋊2Q8 | C42⋊D5 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10⋊Q8 | D10⋊2Q8 | C5×C42.C2 | C4×D5 | C42.C2 | C42 | C4⋊C4 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 12 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{148}D_{10} % in TeX
G:=Group("C4^2.148D10"); // GroupNames label
G:=SmallGroup(320,1361);
// by ID
G=gap.SmallGroup(320,1361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,192,12550]);
// 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^-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
○
≀