metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.126D10, C10.102- (1+4), C10.1092+ (1+4), (C4×Q8)⋊7D5, Q8⋊9(C4×D5), (Q8×C20)⋊9C2, D20⋊25(C2×C4), (C4×D20)⋊37C2, Q8⋊2D5⋊5C4, (Q8×Dic5)⋊9C2, C4⋊C4.325D10, D20⋊8C4⋊17C2, C42⋊D5⋊16C2, C20.72(C22×C4), C10.48(C23×C4), (C2×Q8).202D10, C2.4(D4⋊8D10), (C2×C10).118C24, (C4×C20).170C22, (C2×C20).497C23, D10.20(C22×C4), C22.37(C23×D5), (C2×D20).270C22, C4⋊Dic5.368C22, (Q8×C10).218C22, (C4×Dic5).93C22, Dic5.41(C22×C4), C2.3(Q8.10D10), C5⋊5(C23.33C23), (C2×Dic5).224C23, (C22×D5).187C23, D10⋊C4.163C22, C10.D4.138C22, C4.37(C2×C4×D5), (D5×C4⋊C4)⋊17C2, (C4×D5)⋊5(C2×C4), (C5×Q8)⋊21(C2×C4), C2.29(D5×C22×C4), (C2×C4×D5).79C22, (C2×Q8⋊2D5).6C2, (C5×C4⋊C4).346C22, (C2×C4).654(C22×D5), SmallGroup(320,1246)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 958 in 294 conjugacy classes, 151 normal (22 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×10], C22, C22 [×12], C5, C2×C4, C2×C4 [×6], C2×C4 [×23], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10 [×3], C42 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×6], C20 [×4], D10 [×6], D10 [×6], C2×C10, C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8, C4×Q8, C2×C4○D4, C4×D5 [×12], C4×D5 [×6], D20 [×12], C2×Dic5 [×2], C2×Dic5 [×3], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5 [×3], C23.33C23, C4×Dic5 [×3], C10.D4, C10.D4 [×3], C4⋊Dic5 [×3], D10⋊C4 [×6], C4×C20 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×9], C2×D20 [×3], Q8⋊2D5 [×8], Q8×C10, C42⋊D5 [×3], C4×D20 [×3], D5×C4⋊C4 [×3], D20⋊8C4 [×3], Q8×Dic5, Q8×C20, C2×Q8⋊2D5, C42.126D10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ (1+4), 2- (1+4), C4×D5 [×4], C22×D5 [×7], C23.33C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, Q8.10D10, D4⋊8D10, C42.126D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 155 25 53)(2 54 26 156)(3 157 27 55)(4 56 28 158)(5 159 29 57)(6 58 30 160)(7 141 31 59)(8 60 32 142)(9 143 33 41)(10 42 34 144)(11 145 35 43)(12 44 36 146)(13 147 37 45)(14 46 38 148)(15 149 39 47)(16 48 40 150)(17 151 21 49)(18 50 22 152)(19 153 23 51)(20 52 24 154)(61 108 82 129)(62 130 83 109)(63 110 84 131)(64 132 85 111)(65 112 86 133)(66 134 87 113)(67 114 88 135)(68 136 89 115)(69 116 90 137)(70 138 91 117)(71 118 92 139)(72 140 93 119)(73 120 94 121)(74 122 95 101)(75 102 96 123)(76 124 97 103)(77 104 98 125)(78 126 99 105)(79 106 100 127)(80 128 81 107)
(1 136 35 105)(2 137 36 106)(3 138 37 107)(4 139 38 108)(5 140 39 109)(6 121 40 110)(7 122 21 111)(8 123 22 112)(9 124 23 113)(10 125 24 114)(11 126 25 115)(12 127 26 116)(13 128 27 117)(14 129 28 118)(15 130 29 119)(16 131 30 120)(17 132 31 101)(18 133 32 102)(19 134 33 103)(20 135 34 104)(41 76 153 87)(42 77 154 88)(43 78 155 89)(44 79 156 90)(45 80 157 91)(46 61 158 92)(47 62 159 93)(48 63 160 94)(49 64 141 95)(50 65 142 96)(51 66 143 97)(52 67 144 98)(53 68 145 99)(54 69 146 100)(55 70 147 81)(56 71 148 82)(57 72 149 83)(58 73 150 84)(59 74 151 85)(60 75 152 86)
(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)
(1 109 11 119)(2 118 12 108)(3 107 13 117)(4 116 14 106)(5 105 15 115)(6 114 16 104)(7 103 17 113)(8 112 18 102)(9 101 19 111)(10 110 20 120)(21 134 31 124)(22 123 32 133)(23 132 33 122)(24 121 34 131)(25 130 35 140)(26 139 36 129)(27 128 37 138)(28 137 38 127)(29 126 39 136)(30 135 40 125)(41 74 51 64)(42 63 52 73)(43 72 53 62)(44 61 54 71)(45 70 55 80)(46 79 56 69)(47 68 57 78)(48 77 58 67)(49 66 59 76)(50 75 60 65)(81 147 91 157)(82 156 92 146)(83 145 93 155)(84 154 94 144)(85 143 95 153)(86 152 96 142)(87 141 97 151)(88 150 98 160)(89 159 99 149)(90 148 100 158)
G:=sub<Sym(160)| (1,155,25,53)(2,54,26,156)(3,157,27,55)(4,56,28,158)(5,159,29,57)(6,58,30,160)(7,141,31,59)(8,60,32,142)(9,143,33,41)(10,42,34,144)(11,145,35,43)(12,44,36,146)(13,147,37,45)(14,46,38,148)(15,149,39,47)(16,48,40,150)(17,151,21,49)(18,50,22,152)(19,153,23,51)(20,52,24,154)(61,108,82,129)(62,130,83,109)(63,110,84,131)(64,132,85,111)(65,112,86,133)(66,134,87,113)(67,114,88,135)(68,136,89,115)(69,116,90,137)(70,138,91,117)(71,118,92,139)(72,140,93,119)(73,120,94,121)(74,122,95,101)(75,102,96,123)(76,124,97,103)(77,104,98,125)(78,126,99,105)(79,106,100,127)(80,128,81,107), (1,136,35,105)(2,137,36,106)(3,138,37,107)(4,139,38,108)(5,140,39,109)(6,121,40,110)(7,122,21,111)(8,123,22,112)(9,124,23,113)(10,125,24,114)(11,126,25,115)(12,127,26,116)(13,128,27,117)(14,129,28,118)(15,130,29,119)(16,131,30,120)(17,132,31,101)(18,133,32,102)(19,134,33,103)(20,135,34,104)(41,76,153,87)(42,77,154,88)(43,78,155,89)(44,79,156,90)(45,80,157,91)(46,61,158,92)(47,62,159,93)(48,63,160,94)(49,64,141,95)(50,65,142,96)(51,66,143,97)(52,67,144,98)(53,68,145,99)(54,69,146,100)(55,70,147,81)(56,71,148,82)(57,72,149,83)(58,73,150,84)(59,74,151,85)(60,75,152,86), (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), (1,109,11,119)(2,118,12,108)(3,107,13,117)(4,116,14,106)(5,105,15,115)(6,114,16,104)(7,103,17,113)(8,112,18,102)(9,101,19,111)(10,110,20,120)(21,134,31,124)(22,123,32,133)(23,132,33,122)(24,121,34,131)(25,130,35,140)(26,139,36,129)(27,128,37,138)(28,137,38,127)(29,126,39,136)(30,135,40,125)(41,74,51,64)(42,63,52,73)(43,72,53,62)(44,61,54,71)(45,70,55,80)(46,79,56,69)(47,68,57,78)(48,77,58,67)(49,66,59,76)(50,75,60,65)(81,147,91,157)(82,156,92,146)(83,145,93,155)(84,154,94,144)(85,143,95,153)(86,152,96,142)(87,141,97,151)(88,150,98,160)(89,159,99,149)(90,148,100,158)>;
G:=Group( (1,155,25,53)(2,54,26,156)(3,157,27,55)(4,56,28,158)(5,159,29,57)(6,58,30,160)(7,141,31,59)(8,60,32,142)(9,143,33,41)(10,42,34,144)(11,145,35,43)(12,44,36,146)(13,147,37,45)(14,46,38,148)(15,149,39,47)(16,48,40,150)(17,151,21,49)(18,50,22,152)(19,153,23,51)(20,52,24,154)(61,108,82,129)(62,130,83,109)(63,110,84,131)(64,132,85,111)(65,112,86,133)(66,134,87,113)(67,114,88,135)(68,136,89,115)(69,116,90,137)(70,138,91,117)(71,118,92,139)(72,140,93,119)(73,120,94,121)(74,122,95,101)(75,102,96,123)(76,124,97,103)(77,104,98,125)(78,126,99,105)(79,106,100,127)(80,128,81,107), (1,136,35,105)(2,137,36,106)(3,138,37,107)(4,139,38,108)(5,140,39,109)(6,121,40,110)(7,122,21,111)(8,123,22,112)(9,124,23,113)(10,125,24,114)(11,126,25,115)(12,127,26,116)(13,128,27,117)(14,129,28,118)(15,130,29,119)(16,131,30,120)(17,132,31,101)(18,133,32,102)(19,134,33,103)(20,135,34,104)(41,76,153,87)(42,77,154,88)(43,78,155,89)(44,79,156,90)(45,80,157,91)(46,61,158,92)(47,62,159,93)(48,63,160,94)(49,64,141,95)(50,65,142,96)(51,66,143,97)(52,67,144,98)(53,68,145,99)(54,69,146,100)(55,70,147,81)(56,71,148,82)(57,72,149,83)(58,73,150,84)(59,74,151,85)(60,75,152,86), (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), (1,109,11,119)(2,118,12,108)(3,107,13,117)(4,116,14,106)(5,105,15,115)(6,114,16,104)(7,103,17,113)(8,112,18,102)(9,101,19,111)(10,110,20,120)(21,134,31,124)(22,123,32,133)(23,132,33,122)(24,121,34,131)(25,130,35,140)(26,139,36,129)(27,128,37,138)(28,137,38,127)(29,126,39,136)(30,135,40,125)(41,74,51,64)(42,63,52,73)(43,72,53,62)(44,61,54,71)(45,70,55,80)(46,79,56,69)(47,68,57,78)(48,77,58,67)(49,66,59,76)(50,75,60,65)(81,147,91,157)(82,156,92,146)(83,145,93,155)(84,154,94,144)(85,143,95,153)(86,152,96,142)(87,141,97,151)(88,150,98,160)(89,159,99,149)(90,148,100,158) );
G=PermutationGroup([(1,155,25,53),(2,54,26,156),(3,157,27,55),(4,56,28,158),(5,159,29,57),(6,58,30,160),(7,141,31,59),(8,60,32,142),(9,143,33,41),(10,42,34,144),(11,145,35,43),(12,44,36,146),(13,147,37,45),(14,46,38,148),(15,149,39,47),(16,48,40,150),(17,151,21,49),(18,50,22,152),(19,153,23,51),(20,52,24,154),(61,108,82,129),(62,130,83,109),(63,110,84,131),(64,132,85,111),(65,112,86,133),(66,134,87,113),(67,114,88,135),(68,136,89,115),(69,116,90,137),(70,138,91,117),(71,118,92,139),(72,140,93,119),(73,120,94,121),(74,122,95,101),(75,102,96,123),(76,124,97,103),(77,104,98,125),(78,126,99,105),(79,106,100,127),(80,128,81,107)], [(1,136,35,105),(2,137,36,106),(3,138,37,107),(4,139,38,108),(5,140,39,109),(6,121,40,110),(7,122,21,111),(8,123,22,112),(9,124,23,113),(10,125,24,114),(11,126,25,115),(12,127,26,116),(13,128,27,117),(14,129,28,118),(15,130,29,119),(16,131,30,120),(17,132,31,101),(18,133,32,102),(19,134,33,103),(20,135,34,104),(41,76,153,87),(42,77,154,88),(43,78,155,89),(44,79,156,90),(45,80,157,91),(46,61,158,92),(47,62,159,93),(48,63,160,94),(49,64,141,95),(50,65,142,96),(51,66,143,97),(52,67,144,98),(53,68,145,99),(54,69,146,100),(55,70,147,81),(56,71,148,82),(57,72,149,83),(58,73,150,84),(59,74,151,85),(60,75,152,86)], [(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)], [(1,109,11,119),(2,118,12,108),(3,107,13,117),(4,116,14,106),(5,105,15,115),(6,114,16,104),(7,103,17,113),(8,112,18,102),(9,101,19,111),(10,110,20,120),(21,134,31,124),(22,123,32,133),(23,132,33,122),(24,121,34,131),(25,130,35,140),(26,139,36,129),(27,128,37,138),(28,137,38,127),(29,126,39,136),(30,135,40,125),(41,74,51,64),(42,63,52,73),(43,72,53,62),(44,61,54,71),(45,70,55,80),(46,79,56,69),(47,68,57,78),(48,77,58,67),(49,66,59,76),(50,75,60,65),(81,147,91,157),(82,156,92,146),(83,145,93,155),(84,154,94,144),(85,143,95,153),(86,152,96,142),(87,141,97,151),(88,150,98,160),(89,159,99,149),(90,148,100,158)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 35 | 35 | 29 |
| 0 | 0 | 7 | 3 | 14 | 6 |
| 0 | 0 | 35 | 29 | 3 | 6 |
| 0 | 0 | 14 | 6 | 34 | 38 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 5 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 5 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 22 | 13 | 0 | 0 | 0 | 0 |
| 19 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 22 | 9 | 0 | 0 | 0 | 0 |
| 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 38 | 0 | 0 |
| 0 | 0 | 38 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 38 |
| 0 | 0 | 0 | 0 | 38 | 21 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,38,7,35,14,0,0,35,3,29,6,0,0,35,14,3,34,0,0,29,6,6,38],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,23,1,0,0,0,0,5,18,0,0,0,0,0,0,23,1,0,0,0,0,5,18],[22,19,0,0,0,0,13,0,0,0,0,0,0,0,0,0,35,7,0,0,0,0,35,0,0,0,6,34,0,0,0,0,6,0,0,0],[22,19,0,0,0,0,9,19,0,0,0,0,0,0,20,38,0,0,0,0,38,21,0,0,0,0,0,0,20,38,0,0,0,0,38,21] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4N | 4O | ··· | 4X | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | C4×D5 | 2+ (1+4) | 2- (1+4) | Q8.10D10 | D4⋊8D10 |
| kernel | C42.126D10 | C42⋊D5 | C4×D20 | D5×C4⋊C4 | D20⋊8C4 | Q8×Dic5 | Q8×C20 | C2×Q8⋊2D5 | Q8⋊2D5 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C10 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 2 | 6 | 6 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{126}D_{10} % in TeX
G:=Group("C4^2.126D10"); // GroupNames label
G:=SmallGroup(320,1246);
// by ID
G=gap.SmallGroup(320,1246);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,136,12550]);
// 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=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀