metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.144D10, C10.922- (1+4), C10.1282+ (1+4), C20⋊2Q8⋊31C2, (Q8×Dic5)⋊21C2, (D4×Dic5)⋊32C2, C4.4D4⋊15D5, D10⋊3Q8⋊33C2, (C2×D4).177D10, C20⋊2D4.14C2, C42⋊D5⋊21C2, (C2×C20).82C23, (C2×Q8).140D10, C22⋊C4.37D10, C20.127(C4○D4), C4.39(D4⋊2D5), (C2×C10).226C24, (C4×C20).189C22, C2.52(D4⋊8D10), C23.48(C22×D5), Dic5.5D4⋊42C2, (D4×C10).159C22, C23.D10⋊42C2, C22.D20⋊27C2, C4⋊Dic5.236C22, (C22×C10).56C23, (Q8×C10).130C22, (C22×D5).98C23, C22.247(C23×D5), Dic5.14D4⋊42C2, C23.D5.59C22, C5⋊9(C22.36C24), (C4×Dic5).144C22, (C2×Dic10).40C22, (C2×Dic5).116C23, C10.D4.49C22, C2.53(D4.10D10), D10⋊C4.111C22, (C22×Dic5).146C22, C10.94(C2×C4○D4), (C5×C4.4D4)⋊18C2, C2.58(C2×D4⋊2D5), (C2×C4×D5).131C22, (C2×C4).199(C22×D5), (C2×C5⋊D4).64C22, (C5×C22⋊C4).68C22, SmallGroup(320,1354)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 734 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 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], 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, C4.4D4 [×2], C42⋊2C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10 [×2], C22.36C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C20⋊2Q8, C42⋊D5, Dic5.14D4 [×2], C23.D10 [×2], Dic5.5D4 [×2], C22.D20 [×2], D4×Dic5, C20⋊2D4, Q8×Dic5, D10⋊3Q8, C5×C4.4D4, C42.144D10
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, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D4⋊8D10, D4.10D10, C42.144D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, 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)(11 131 56 91)(12 92 57 132)(13 133 58 93)(14 94 59 134)(15 135 60 95)(16 96 51 136)(17 137 52 97)(18 98 53 138)(19 139 54 99)(20 100 55 140)(21 90 149 63)(22 64 150 81)(23 82 141 65)(24 66 142 83)(25 84 143 67)(26 68 144 85)(27 86 145 69)(28 70 146 87)(29 88 147 61)(30 62 148 89)(41 154 74 122)(42 123 75 155)(43 156 76 124)(44 125 77 157)(45 158 78 126)(46 127 79 159)(47 160 80 128)(48 129 71 151)(49 152 72 130)(50 121 73 153)
(1 73 20 85)(2 69 11 41)(3 75 12 87)(4 61 13 43)(5 77 14 89)(6 63 15 45)(7 79 16 81)(8 65 17 47)(9 71 18 83)(10 67 19 49)(21 135 158 118)(22 104 159 96)(23 137 160 120)(24 106 151 98)(25 139 152 112)(26 108 153 100)(27 131 154 114)(28 110 155 92)(29 133 156 116)(30 102 157 94)(31 48 53 66)(32 84 54 72)(33 50 55 68)(34 86 56 74)(35 42 57 70)(36 88 58 76)(37 44 59 62)(38 90 60 78)(39 46 51 64)(40 82 52 80)(91 122 109 145)(93 124 101 147)(95 126 103 149)(97 128 105 141)(99 130 107 143)(111 129 138 142)(113 121 140 144)(115 123 132 146)(117 125 134 148)(119 127 136 150)
(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 20 19)(2 18 11 9)(3 8 12 17)(4 16 13 7)(5 6 14 15)(21 125 158 148)(22 147 159 124)(23 123 160 146)(24 145 151 122)(25 121 152 144)(26 143 153 130)(27 129 154 142)(28 141 155 128)(29 127 156 150)(30 149 157 126)(31 34 53 56)(32 55 54 33)(35 40 57 52)(36 51 58 39)(37 38 59 60)(41 83 69 71)(42 80 70 82)(43 81 61 79)(44 78 62 90)(45 89 63 77)(46 76 64 88)(47 87 65 75)(48 74 66 86)(49 85 67 73)(50 72 68 84)(91 106 109 98)(92 97 110 105)(93 104 101 96)(94 95 102 103)(99 108 107 100)(111 114 138 131)(112 140 139 113)(115 120 132 137)(116 136 133 119)(117 118 134 135)
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)(11,131,56,91)(12,92,57,132)(13,133,58,93)(14,94,59,134)(15,135,60,95)(16,96,51,136)(17,137,52,97)(18,98,53,138)(19,139,54,99)(20,100,55,140)(21,90,149,63)(22,64,150,81)(23,82,141,65)(24,66,142,83)(25,84,143,67)(26,68,144,85)(27,86,145,69)(28,70,146,87)(29,88,147,61)(30,62,148,89)(41,154,74,122)(42,123,75,155)(43,156,76,124)(44,125,77,157)(45,158,78,126)(46,127,79,159)(47,160,80,128)(48,129,71,151)(49,152,72,130)(50,121,73,153), (1,73,20,85)(2,69,11,41)(3,75,12,87)(4,61,13,43)(5,77,14,89)(6,63,15,45)(7,79,16,81)(8,65,17,47)(9,71,18,83)(10,67,19,49)(21,135,158,118)(22,104,159,96)(23,137,160,120)(24,106,151,98)(25,139,152,112)(26,108,153,100)(27,131,154,114)(28,110,155,92)(29,133,156,116)(30,102,157,94)(31,48,53,66)(32,84,54,72)(33,50,55,68)(34,86,56,74)(35,42,57,70)(36,88,58,76)(37,44,59,62)(38,90,60,78)(39,46,51,64)(40,82,52,80)(91,122,109,145)(93,124,101,147)(95,126,103,149)(97,128,105,141)(99,130,107,143)(111,129,138,142)(113,121,140,144)(115,123,132,146)(117,125,134,148)(119,127,136,150), (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), (1,10,20,19)(2,18,11,9)(3,8,12,17)(4,16,13,7)(5,6,14,15)(21,125,158,148)(22,147,159,124)(23,123,160,146)(24,145,151,122)(25,121,152,144)(26,143,153,130)(27,129,154,142)(28,141,155,128)(29,127,156,150)(30,149,157,126)(31,34,53,56)(32,55,54,33)(35,40,57,52)(36,51,58,39)(37,38,59,60)(41,83,69,71)(42,80,70,82)(43,81,61,79)(44,78,62,90)(45,89,63,77)(46,76,64,88)(47,87,65,75)(48,74,66,86)(49,85,67,73)(50,72,68,84)(91,106,109,98)(92,97,110,105)(93,104,101,96)(94,95,102,103)(99,108,107,100)(111,114,138,131)(112,140,139,113)(115,120,132,137)(116,136,133,119)(117,118,134,135)>;
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)(11,131,56,91)(12,92,57,132)(13,133,58,93)(14,94,59,134)(15,135,60,95)(16,96,51,136)(17,137,52,97)(18,98,53,138)(19,139,54,99)(20,100,55,140)(21,90,149,63)(22,64,150,81)(23,82,141,65)(24,66,142,83)(25,84,143,67)(26,68,144,85)(27,86,145,69)(28,70,146,87)(29,88,147,61)(30,62,148,89)(41,154,74,122)(42,123,75,155)(43,156,76,124)(44,125,77,157)(45,158,78,126)(46,127,79,159)(47,160,80,128)(48,129,71,151)(49,152,72,130)(50,121,73,153), (1,73,20,85)(2,69,11,41)(3,75,12,87)(4,61,13,43)(5,77,14,89)(6,63,15,45)(7,79,16,81)(8,65,17,47)(9,71,18,83)(10,67,19,49)(21,135,158,118)(22,104,159,96)(23,137,160,120)(24,106,151,98)(25,139,152,112)(26,108,153,100)(27,131,154,114)(28,110,155,92)(29,133,156,116)(30,102,157,94)(31,48,53,66)(32,84,54,72)(33,50,55,68)(34,86,56,74)(35,42,57,70)(36,88,58,76)(37,44,59,62)(38,90,60,78)(39,46,51,64)(40,82,52,80)(91,122,109,145)(93,124,101,147)(95,126,103,149)(97,128,105,141)(99,130,107,143)(111,129,138,142)(113,121,140,144)(115,123,132,146)(117,125,134,148)(119,127,136,150), (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), (1,10,20,19)(2,18,11,9)(3,8,12,17)(4,16,13,7)(5,6,14,15)(21,125,158,148)(22,147,159,124)(23,123,160,146)(24,145,151,122)(25,121,152,144)(26,143,153,130)(27,129,154,142)(28,141,155,128)(29,127,156,150)(30,149,157,126)(31,34,53,56)(32,55,54,33)(35,40,57,52)(36,51,58,39)(37,38,59,60)(41,83,69,71)(42,80,70,82)(43,81,61,79)(44,78,62,90)(45,89,63,77)(46,76,64,88)(47,87,65,75)(48,74,66,86)(49,85,67,73)(50,72,68,84)(91,106,109,98)(92,97,110,105)(93,104,101,96)(94,95,102,103)(99,108,107,100)(111,114,138,131)(112,140,139,113)(115,120,132,137)(116,136,133,119)(117,118,134,135) );
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),(11,131,56,91),(12,92,57,132),(13,133,58,93),(14,94,59,134),(15,135,60,95),(16,96,51,136),(17,137,52,97),(18,98,53,138),(19,139,54,99),(20,100,55,140),(21,90,149,63),(22,64,150,81),(23,82,141,65),(24,66,142,83),(25,84,143,67),(26,68,144,85),(27,86,145,69),(28,70,146,87),(29,88,147,61),(30,62,148,89),(41,154,74,122),(42,123,75,155),(43,156,76,124),(44,125,77,157),(45,158,78,126),(46,127,79,159),(47,160,80,128),(48,129,71,151),(49,152,72,130),(50,121,73,153)], [(1,73,20,85),(2,69,11,41),(3,75,12,87),(4,61,13,43),(5,77,14,89),(6,63,15,45),(7,79,16,81),(8,65,17,47),(9,71,18,83),(10,67,19,49),(21,135,158,118),(22,104,159,96),(23,137,160,120),(24,106,151,98),(25,139,152,112),(26,108,153,100),(27,131,154,114),(28,110,155,92),(29,133,156,116),(30,102,157,94),(31,48,53,66),(32,84,54,72),(33,50,55,68),(34,86,56,74),(35,42,57,70),(36,88,58,76),(37,44,59,62),(38,90,60,78),(39,46,51,64),(40,82,52,80),(91,122,109,145),(93,124,101,147),(95,126,103,149),(97,128,105,141),(99,130,107,143),(111,129,138,142),(113,121,140,144),(115,123,132,146),(117,125,134,148),(119,127,136,150)], [(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)], [(1,10,20,19),(2,18,11,9),(3,8,12,17),(4,16,13,7),(5,6,14,15),(21,125,158,148),(22,147,159,124),(23,123,160,146),(24,145,151,122),(25,121,152,144),(26,143,153,130),(27,129,154,142),(28,141,155,128),(29,127,156,150),(30,149,157,126),(31,34,53,56),(32,55,54,33),(35,40,57,52),(36,51,58,39),(37,38,59,60),(41,83,69,71),(42,80,70,82),(43,81,61,79),(44,78,62,90),(45,89,63,77),(46,76,64,88),(47,87,65,75),(48,74,66,86),(49,85,67,73),(50,72,68,84),(91,106,109,98),(92,97,110,105),(93,104,101,96),(94,95,102,103),(99,108,107,100),(111,114,138,131),(112,140,139,113),(115,120,132,137),(116,136,133,119),(117,118,134,135)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 29 | 0 |
| 0 | 0 | 0 | 26 | 0 | 29 |
| 0 | 0 | 12 | 0 | 15 | 0 |
| 0 | 0 | 0 | 12 | 0 | 15 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 12 | 24 | 26 |
| 0 | 0 | 19 | 11 | 7 | 17 |
| 0 | 0 | 17 | 15 | 11 | 29 |
| 0 | 0 | 34 | 24 | 22 | 30 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 7 |
| 0 | 0 | 0 | 0 | 36 | 6 |
| 0 | 0 | 35 | 7 | 0 | 0 |
| 0 | 0 | 36 | 6 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,26,0,12,0,0,0,0,26,0,12,0,0,29,0,15,0,0,0,0,29,0,15],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,30,19,17,34,0,0,12,11,15,24,0,0,24,7,11,22,0,0,26,17,29,30],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,35,0,0,0,0,7,6,0,0,0,35,0,0,0,0,7,6,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,35,36,0,0,0,0,7,6,0,0,35,36,0,0,0,0,7,6,0,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2+ (1+4) | 2- (1+4) | D4⋊2D5 | D4⋊8D10 | D4.10D10 |
| kernel | C42.144D10 | C20⋊2Q8 | C42⋊D5 | Dic5.14D4 | C23.D10 | Dic5.5D4 | C22.D20 | D4×Dic5 | C20⋊2D4 | Q8×Dic5 | D10⋊3Q8 | C5×C4.4D4 | C4.4D4 | C20 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{144}D_{10} % in TeX
G:=Group("C4^2.144D10"); // GroupNames label
G:=SmallGroup(320,1354);
// by ID
G=gap.SmallGroup(320,1354);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,409,12550]);
// 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,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀