metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.115D10, C10.222+ (1+4), C10.622- (1+4), (C4×D4)⋊23D5, C20⋊Q8⋊16C2, (D4×C20)⋊25C2, C4⋊C4.286D10, D10⋊2Q8⋊16C2, (C4×Dic10)⋊35C2, (C2×D4).222D10, C4.45(C4○D20), C20⋊2D4.11C2, C42⋊D5⋊14C2, C20.112(C4○D4), C20.48D4⋊12C2, C20.17D4⋊10C2, (C2×C20).163C23, (C4×C20).159C22, (C2×C10).105C24, C22⋊C4.117D10, (C22×C4).213D10, C23.D10⋊9C2, C4⋊Dic5.40C22, C2.23(D4⋊6D10), Dic5.5D4⋊10C2, (D4×C10).264C22, C23.23D10⋊3C2, (C22×C20).82C22, (C2×Dic5).46C23, (C4×Dic5).85C22, (C22×D5).39C23, C22.130(C23×D5), C23.102(C22×D5), C23.D5.15C22, (C22×C10).175C23, C5⋊2(C22.36C24), (C2×Dic10).29C22, C2.19(D4.10D10), D10⋊C4.123C22, C10.D4.135C22, C2.54(C2×C4○D20), C10.47(C2×C4○D4), (C2×C4×D5).76C22, (C5×C4⋊C4).333C22, (C2×C4).287(C22×D5), (C2×C5⋊D4).18C22, (C5×C22⋊C4).128C22, SmallGroup(320,1233)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 718 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, D5, C10 [×3], C10 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×2], C2×Q8 [×3], Dic5 [×7], C20 [×2], C20 [×4], D10 [×3], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic10 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×C10 [×2], C22.36C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C42⋊D5, C23.D10 [×2], Dic5.5D4 [×2], C20⋊Q8, D10⋊2Q8, C20.48D4 [×2], C23.23D10 [×2], C20.17D4, C20⋊2D4, D4×C20, C42.115D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.36C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊6D10, D4.10D10, C42.115D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >
►On 160 points
Generators in S160
(1 108 33 113)(2 114 34 109)(3 110 35 115)(4 116 36 101)(5 102 37 117)(6 118 38 103)(7 104 39 119)(8 120 40 105)(9 106 31 111)(10 112 32 107)(11 124 49 99)(12 100 50 125)(13 126 41 91)(14 92 42 127)(15 128 43 93)(16 94 44 129)(17 130 45 95)(18 96 46 121)(19 122 47 97)(20 98 48 123)(21 90 135 65)(22 66 136 81)(23 82 137 67)(24 68 138 83)(25 84 139 69)(26 70 140 85)(27 86 131 61)(28 62 132 87)(29 88 133 63)(30 64 134 89)(51 152 72 147)(52 148 73 153)(53 154 74 149)(54 150 75 155)(55 156 76 141)(56 142 77 157)(57 158 78 143)(58 144 79 159)(59 160 80 145)(60 146 71 151)
(1 70 12 52)(2 61 13 53)(3 62 14 54)(4 63 15 55)(5 64 16 56)(6 65 17 57)(7 66 18 58)(8 67 19 59)(9 68 20 60)(10 69 11 51)(21 130 158 118)(22 121 159 119)(23 122 160 120)(24 123 151 111)(25 124 152 112)(26 125 153 113)(27 126 154 114)(28 127 155 115)(29 128 156 116)(30 129 157 117)(31 83 48 71)(32 84 49 72)(33 85 50 73)(34 86 41 74)(35 87 42 75)(36 88 43 76)(37 89 44 77)(38 90 45 78)(39 81 46 79)(40 82 47 80)(91 149 109 131)(92 150 110 132)(93 141 101 133)(94 142 102 134)(95 143 103 135)(96 144 104 136)(97 145 105 137)(98 146 106 138)(99 147 107 139)(100 148 108 140)
(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 12 11)(2 20 13 9)(3 8 14 19)(4 18 15 7)(5 6 16 17)(21 142 158 134)(22 133 159 141)(23 150 160 132)(24 131 151 149)(25 148 152 140)(26 139 153 147)(27 146 154 138)(28 137 155 145)(29 144 156 136)(30 135 157 143)(31 34 48 41)(32 50 49 33)(35 40 42 47)(36 46 43 39)(37 38 44 45)(51 85 69 73)(52 72 70 84)(53 83 61 71)(54 80 62 82)(55 81 63 79)(56 78 64 90)(57 89 65 77)(58 76 66 88)(59 87 67 75)(60 74 68 86)(91 106 109 98)(92 97 110 105)(93 104 101 96)(94 95 102 103)(99 108 107 100)(111 114 123 126)(112 125 124 113)(115 120 127 122)(116 121 128 119)(117 118 129 130)
G:=sub<Sym(160)| (1,108,33,113)(2,114,34,109)(3,110,35,115)(4,116,36,101)(5,102,37,117)(6,118,38,103)(7,104,39,119)(8,120,40,105)(9,106,31,111)(10,112,32,107)(11,124,49,99)(12,100,50,125)(13,126,41,91)(14,92,42,127)(15,128,43,93)(16,94,44,129)(17,130,45,95)(18,96,46,121)(19,122,47,97)(20,98,48,123)(21,90,135,65)(22,66,136,81)(23,82,137,67)(24,68,138,83)(25,84,139,69)(26,70,140,85)(27,86,131,61)(28,62,132,87)(29,88,133,63)(30,64,134,89)(51,152,72,147)(52,148,73,153)(53,154,74,149)(54,150,75,155)(55,156,76,141)(56,142,77,157)(57,158,78,143)(58,144,79,159)(59,160,80,145)(60,146,71,151), (1,70,12,52)(2,61,13,53)(3,62,14,54)(4,63,15,55)(5,64,16,56)(6,65,17,57)(7,66,18,58)(8,67,19,59)(9,68,20,60)(10,69,11,51)(21,130,158,118)(22,121,159,119)(23,122,160,120)(24,123,151,111)(25,124,152,112)(26,125,153,113)(27,126,154,114)(28,127,155,115)(29,128,156,116)(30,129,157,117)(31,83,48,71)(32,84,49,72)(33,85,50,73)(34,86,41,74)(35,87,42,75)(36,88,43,76)(37,89,44,77)(38,90,45,78)(39,81,46,79)(40,82,47,80)(91,149,109,131)(92,150,110,132)(93,141,101,133)(94,142,102,134)(95,143,103,135)(96,144,104,136)(97,145,105,137)(98,146,106,138)(99,147,107,139)(100,148,108,140), (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), (1,10,12,11)(2,20,13,9)(3,8,14,19)(4,18,15,7)(5,6,16,17)(21,142,158,134)(22,133,159,141)(23,150,160,132)(24,131,151,149)(25,148,152,140)(26,139,153,147)(27,146,154,138)(28,137,155,145)(29,144,156,136)(30,135,157,143)(31,34,48,41)(32,50,49,33)(35,40,42,47)(36,46,43,39)(37,38,44,45)(51,85,69,73)(52,72,70,84)(53,83,61,71)(54,80,62,82)(55,81,63,79)(56,78,64,90)(57,89,65,77)(58,76,66,88)(59,87,67,75)(60,74,68,86)(91,106,109,98)(92,97,110,105)(93,104,101,96)(94,95,102,103)(99,108,107,100)(111,114,123,126)(112,125,124,113)(115,120,127,122)(116,121,128,119)(117,118,129,130)>;
G:=Group( (1,108,33,113)(2,114,34,109)(3,110,35,115)(4,116,36,101)(5,102,37,117)(6,118,38,103)(7,104,39,119)(8,120,40,105)(9,106,31,111)(10,112,32,107)(11,124,49,99)(12,100,50,125)(13,126,41,91)(14,92,42,127)(15,128,43,93)(16,94,44,129)(17,130,45,95)(18,96,46,121)(19,122,47,97)(20,98,48,123)(21,90,135,65)(22,66,136,81)(23,82,137,67)(24,68,138,83)(25,84,139,69)(26,70,140,85)(27,86,131,61)(28,62,132,87)(29,88,133,63)(30,64,134,89)(51,152,72,147)(52,148,73,153)(53,154,74,149)(54,150,75,155)(55,156,76,141)(56,142,77,157)(57,158,78,143)(58,144,79,159)(59,160,80,145)(60,146,71,151), (1,70,12,52)(2,61,13,53)(3,62,14,54)(4,63,15,55)(5,64,16,56)(6,65,17,57)(7,66,18,58)(8,67,19,59)(9,68,20,60)(10,69,11,51)(21,130,158,118)(22,121,159,119)(23,122,160,120)(24,123,151,111)(25,124,152,112)(26,125,153,113)(27,126,154,114)(28,127,155,115)(29,128,156,116)(30,129,157,117)(31,83,48,71)(32,84,49,72)(33,85,50,73)(34,86,41,74)(35,87,42,75)(36,88,43,76)(37,89,44,77)(38,90,45,78)(39,81,46,79)(40,82,47,80)(91,149,109,131)(92,150,110,132)(93,141,101,133)(94,142,102,134)(95,143,103,135)(96,144,104,136)(97,145,105,137)(98,146,106,138)(99,147,107,139)(100,148,108,140), (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), (1,10,12,11)(2,20,13,9)(3,8,14,19)(4,18,15,7)(5,6,16,17)(21,142,158,134)(22,133,159,141)(23,150,160,132)(24,131,151,149)(25,148,152,140)(26,139,153,147)(27,146,154,138)(28,137,155,145)(29,144,156,136)(30,135,157,143)(31,34,48,41)(32,50,49,33)(35,40,42,47)(36,46,43,39)(37,38,44,45)(51,85,69,73)(52,72,70,84)(53,83,61,71)(54,80,62,82)(55,81,63,79)(56,78,64,90)(57,89,65,77)(58,76,66,88)(59,87,67,75)(60,74,68,86)(91,106,109,98)(92,97,110,105)(93,104,101,96)(94,95,102,103)(99,108,107,100)(111,114,123,126)(112,125,124,113)(115,120,127,122)(116,121,128,119)(117,118,129,130) );
G=PermutationGroup([(1,108,33,113),(2,114,34,109),(3,110,35,115),(4,116,36,101),(5,102,37,117),(6,118,38,103),(7,104,39,119),(8,120,40,105),(9,106,31,111),(10,112,32,107),(11,124,49,99),(12,100,50,125),(13,126,41,91),(14,92,42,127),(15,128,43,93),(16,94,44,129),(17,130,45,95),(18,96,46,121),(19,122,47,97),(20,98,48,123),(21,90,135,65),(22,66,136,81),(23,82,137,67),(24,68,138,83),(25,84,139,69),(26,70,140,85),(27,86,131,61),(28,62,132,87),(29,88,133,63),(30,64,134,89),(51,152,72,147),(52,148,73,153),(53,154,74,149),(54,150,75,155),(55,156,76,141),(56,142,77,157),(57,158,78,143),(58,144,79,159),(59,160,80,145),(60,146,71,151)], [(1,70,12,52),(2,61,13,53),(3,62,14,54),(4,63,15,55),(5,64,16,56),(6,65,17,57),(7,66,18,58),(8,67,19,59),(9,68,20,60),(10,69,11,51),(21,130,158,118),(22,121,159,119),(23,122,160,120),(24,123,151,111),(25,124,152,112),(26,125,153,113),(27,126,154,114),(28,127,155,115),(29,128,156,116),(30,129,157,117),(31,83,48,71),(32,84,49,72),(33,85,50,73),(34,86,41,74),(35,87,42,75),(36,88,43,76),(37,89,44,77),(38,90,45,78),(39,81,46,79),(40,82,47,80),(91,149,109,131),(92,150,110,132),(93,141,101,133),(94,142,102,134),(95,143,103,135),(96,144,104,136),(97,145,105,137),(98,146,106,138),(99,147,107,139),(100,148,108,140)], [(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)], [(1,10,12,11),(2,20,13,9),(3,8,14,19),(4,18,15,7),(5,6,16,17),(21,142,158,134),(22,133,159,141),(23,150,160,132),(24,131,151,149),(25,148,152,140),(26,139,153,147),(27,146,154,138),(28,137,155,145),(29,144,156,136),(30,135,157,143),(31,34,48,41),(32,50,49,33),(35,40,42,47),(36,46,43,39),(37,38,44,45),(51,85,69,73),(52,72,70,84),(53,83,61,71),(54,80,62,82),(55,81,63,79),(56,78,64,90),(57,89,65,77),(58,76,66,88),(59,87,67,75),(60,74,68,86),(91,106,109,98),(92,97,110,105),(93,104,101,96),(94,95,102,103),(99,108,107,100),(111,114,123,126),(112,125,124,113),(115,120,127,122),(116,121,128,119),(117,118,129,130)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 31 | 22 | 21 |
| 0 | 0 | 10 | 15 | 20 | 19 |
| 0 | 0 | 4 | 10 | 4 | 10 |
| 0 | 0 | 31 | 26 | 31 | 26 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 28 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 28 |
| 0 | 0 | 0 | 0 | 13 | 39 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 2 | 12 |
| 0 | 0 | 35 | 6 | 29 | 12 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 39 | 0 |
| 0 | 0 | 6 | 1 | 12 | 2 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 35 | 40 | 35 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,10,4,31,0,0,31,15,10,26,0,0,22,20,4,31,0,0,21,19,10,26],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,0,0,0,0,2,13,0,0,0,0,28,39],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,35,0,0,0,0,6,6,0,0,0,0,2,29,40,6,0,0,12,12,35,35],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,6,1,35,0,0,0,1,0,40,0,0,39,12,1,35,0,0,0,2,0,40] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D4.10D10 |
| kernel | C42.115D10 | C4×Dic10 | C42⋊D5 | C23.D10 | Dic5.5D4 | C20⋊Q8 | D10⋊2Q8 | C20.48D4 | C23.23D10 | C20.17D4 | C20⋊2D4 | D4×C20 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{115}D_{10} % in TeX
G:=Group("C4^2.115D10"); // GroupNames label
G:=SmallGroup(320,1233);
// by ID
G=gap.SmallGroup(320,1233);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^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=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀