metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.100D10, C10.1002+ (1+4), (C4×D20)⋊12C2, C4⋊D20⋊4C2, D10⋊D4⋊5C2, C20⋊7D4⋊42C2, C4⋊C4.275D10, C20.6Q8⋊5C2, C42⋊C2⋊19D5, (C4×C20).30C22, (C2×C10).79C24, D10.13D4⋊5C2, C4.121(C4○D20), C20.237(C4○D4), (C2×C20).152C23, C22⋊C4.103D10, (C2×D20).27C22, (C22×C4).200D10, C2.12(D4⋊8D10), C23.90(C22×D5), C4⋊Dic5.294C22, (C2×Dic5).32C23, C10.D4.4C22, (C22×D5).27C23, C22.108(C23×D5), D10⋊C4.64C22, (C22×C20).309C22, (C22×C10).149C23, C5⋊1(C22.34C24), C10.35(C2×C4○D4), C2.38(C2×C4○D20), (C2×C4×D5).247C22, (C5×C42⋊C2)⋊21C2, (C5×C4⋊C4).315C22, (C2×C4).280(C22×D5), (C2×C5⋊D4).12C22, (C5×C22⋊C4).118C22, SmallGroup(320,1207)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1022 in 240 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], C23, C23 [×4], D5 [×4], C10, C10 [×2], C10, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4 [×10], Dic5 [×4], C20 [×2], C20 [×5], D10 [×12], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D5 [×4], D20 [×8], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5 [×4], C22×C10, C22.34C24, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5 [×4], C2×D20 [×6], C2×C5⋊D4 [×4], C22×C20, C20.6Q8, C4×D20 [×2], C4⋊D20, D10⋊D4 [×4], D10.13D4 [×4], C20⋊7D4 [×2], C5×C42⋊C2, C42.100D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D5 [×7], C22.34C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊8D10 [×2], C42.100D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c9 >
►On 160 points
Generators in S160
(1 65 11 75)(2 66 12 76)(3 67 13 77)(4 68 14 78)(5 69 15 79)(6 70 16 80)(7 71 17 61)(8 72 18 62)(9 73 19 63)(10 74 20 64)(21 134 31 124)(22 135 32 125)(23 136 33 126)(24 137 34 127)(25 138 35 128)(26 139 36 129)(27 140 37 130)(28 121 38 131)(29 122 39 132)(30 123 40 133)(41 154 51 144)(42 155 52 145)(43 156 53 146)(44 157 54 147)(45 158 55 148)(46 159 56 149)(47 160 57 150)(48 141 58 151)(49 142 59 152)(50 143 60 153)(81 110 91 120)(82 111 92 101)(83 112 93 102)(84 113 94 103)(85 114 95 104)(86 115 96 105)(87 116 97 106)(88 117 98 107)(89 118 99 108)(90 119 100 109)
(1 115 22 45)(2 106 23 56)(3 117 24 47)(4 108 25 58)(5 119 26 49)(6 110 27 60)(7 101 28 51)(8 112 29 42)(9 103 30 53)(10 114 31 44)(11 105 32 55)(12 116 33 46)(13 107 34 57)(14 118 35 48)(15 109 36 59)(16 120 37 50)(17 111 38 41)(18 102 39 52)(19 113 40 43)(20 104 21 54)(61 92 131 154)(62 83 132 145)(63 94 133 156)(64 85 134 147)(65 96 135 158)(66 87 136 149)(67 98 137 160)(68 89 138 151)(69 100 139 142)(70 91 140 153)(71 82 121 144)(72 93 122 155)(73 84 123 146)(74 95 124 157)(75 86 125 148)(76 97 126 159)(77 88 127 150)(78 99 128 141)(79 90 129 152)(80 81 130 143)
(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 64 32 124)(2 123 33 63)(3 62 34 122)(4 121 35 61)(5 80 36 140)(6 139 37 79)(7 78 38 138)(8 137 39 77)(9 76 40 136)(10 135 21 75)(11 74 22 134)(12 133 23 73)(13 72 24 132)(14 131 25 71)(15 70 26 130)(16 129 27 69)(17 68 28 128)(18 127 29 67)(19 66 30 126)(20 125 31 65)(41 141 101 89)(42 88 102 160)(43 159 103 87)(44 86 104 158)(45 157 105 85)(46 84 106 156)(47 155 107 83)(48 82 108 154)(49 153 109 81)(50 100 110 152)(51 151 111 99)(52 98 112 150)(53 149 113 97)(54 96 114 148)(55 147 115 95)(56 94 116 146)(57 145 117 93)(58 92 118 144)(59 143 119 91)(60 90 120 142)
G:=sub<Sym(160)| (1,65,11,75)(2,66,12,76)(3,67,13,77)(4,68,14,78)(5,69,15,79)(6,70,16,80)(7,71,17,61)(8,72,18,62)(9,73,19,63)(10,74,20,64)(21,134,31,124)(22,135,32,125)(23,136,33,126)(24,137,34,127)(25,138,35,128)(26,139,36,129)(27,140,37,130)(28,121,38,131)(29,122,39,132)(30,123,40,133)(41,154,51,144)(42,155,52,145)(43,156,53,146)(44,157,54,147)(45,158,55,148)(46,159,56,149)(47,160,57,150)(48,141,58,151)(49,142,59,152)(50,143,60,153)(81,110,91,120)(82,111,92,101)(83,112,93,102)(84,113,94,103)(85,114,95,104)(86,115,96,105)(87,116,97,106)(88,117,98,107)(89,118,99,108)(90,119,100,109), (1,115,22,45)(2,106,23,56)(3,117,24,47)(4,108,25,58)(5,119,26,49)(6,110,27,60)(7,101,28,51)(8,112,29,42)(9,103,30,53)(10,114,31,44)(11,105,32,55)(12,116,33,46)(13,107,34,57)(14,118,35,48)(15,109,36,59)(16,120,37,50)(17,111,38,41)(18,102,39,52)(19,113,40,43)(20,104,21,54)(61,92,131,154)(62,83,132,145)(63,94,133,156)(64,85,134,147)(65,96,135,158)(66,87,136,149)(67,98,137,160)(68,89,138,151)(69,100,139,142)(70,91,140,153)(71,82,121,144)(72,93,122,155)(73,84,123,146)(74,95,124,157)(75,86,125,148)(76,97,126,159)(77,88,127,150)(78,99,128,141)(79,90,129,152)(80,81,130,143), (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), (1,64,32,124)(2,123,33,63)(3,62,34,122)(4,121,35,61)(5,80,36,140)(6,139,37,79)(7,78,38,138)(8,137,39,77)(9,76,40,136)(10,135,21,75)(11,74,22,134)(12,133,23,73)(13,72,24,132)(14,131,25,71)(15,70,26,130)(16,129,27,69)(17,68,28,128)(18,127,29,67)(19,66,30,126)(20,125,31,65)(41,141,101,89)(42,88,102,160)(43,159,103,87)(44,86,104,158)(45,157,105,85)(46,84,106,156)(47,155,107,83)(48,82,108,154)(49,153,109,81)(50,100,110,152)(51,151,111,99)(52,98,112,150)(53,149,113,97)(54,96,114,148)(55,147,115,95)(56,94,116,146)(57,145,117,93)(58,92,118,144)(59,143,119,91)(60,90,120,142)>;
G:=Group( (1,65,11,75)(2,66,12,76)(3,67,13,77)(4,68,14,78)(5,69,15,79)(6,70,16,80)(7,71,17,61)(8,72,18,62)(9,73,19,63)(10,74,20,64)(21,134,31,124)(22,135,32,125)(23,136,33,126)(24,137,34,127)(25,138,35,128)(26,139,36,129)(27,140,37,130)(28,121,38,131)(29,122,39,132)(30,123,40,133)(41,154,51,144)(42,155,52,145)(43,156,53,146)(44,157,54,147)(45,158,55,148)(46,159,56,149)(47,160,57,150)(48,141,58,151)(49,142,59,152)(50,143,60,153)(81,110,91,120)(82,111,92,101)(83,112,93,102)(84,113,94,103)(85,114,95,104)(86,115,96,105)(87,116,97,106)(88,117,98,107)(89,118,99,108)(90,119,100,109), (1,115,22,45)(2,106,23,56)(3,117,24,47)(4,108,25,58)(5,119,26,49)(6,110,27,60)(7,101,28,51)(8,112,29,42)(9,103,30,53)(10,114,31,44)(11,105,32,55)(12,116,33,46)(13,107,34,57)(14,118,35,48)(15,109,36,59)(16,120,37,50)(17,111,38,41)(18,102,39,52)(19,113,40,43)(20,104,21,54)(61,92,131,154)(62,83,132,145)(63,94,133,156)(64,85,134,147)(65,96,135,158)(66,87,136,149)(67,98,137,160)(68,89,138,151)(69,100,139,142)(70,91,140,153)(71,82,121,144)(72,93,122,155)(73,84,123,146)(74,95,124,157)(75,86,125,148)(76,97,126,159)(77,88,127,150)(78,99,128,141)(79,90,129,152)(80,81,130,143), (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), (1,64,32,124)(2,123,33,63)(3,62,34,122)(4,121,35,61)(5,80,36,140)(6,139,37,79)(7,78,38,138)(8,137,39,77)(9,76,40,136)(10,135,21,75)(11,74,22,134)(12,133,23,73)(13,72,24,132)(14,131,25,71)(15,70,26,130)(16,129,27,69)(17,68,28,128)(18,127,29,67)(19,66,30,126)(20,125,31,65)(41,141,101,89)(42,88,102,160)(43,159,103,87)(44,86,104,158)(45,157,105,85)(46,84,106,156)(47,155,107,83)(48,82,108,154)(49,153,109,81)(50,100,110,152)(51,151,111,99)(52,98,112,150)(53,149,113,97)(54,96,114,148)(55,147,115,95)(56,94,116,146)(57,145,117,93)(58,92,118,144)(59,143,119,91)(60,90,120,142) );
G=PermutationGroup([(1,65,11,75),(2,66,12,76),(3,67,13,77),(4,68,14,78),(5,69,15,79),(6,70,16,80),(7,71,17,61),(8,72,18,62),(9,73,19,63),(10,74,20,64),(21,134,31,124),(22,135,32,125),(23,136,33,126),(24,137,34,127),(25,138,35,128),(26,139,36,129),(27,140,37,130),(28,121,38,131),(29,122,39,132),(30,123,40,133),(41,154,51,144),(42,155,52,145),(43,156,53,146),(44,157,54,147),(45,158,55,148),(46,159,56,149),(47,160,57,150),(48,141,58,151),(49,142,59,152),(50,143,60,153),(81,110,91,120),(82,111,92,101),(83,112,93,102),(84,113,94,103),(85,114,95,104),(86,115,96,105),(87,116,97,106),(88,117,98,107),(89,118,99,108),(90,119,100,109)], [(1,115,22,45),(2,106,23,56),(3,117,24,47),(4,108,25,58),(5,119,26,49),(6,110,27,60),(7,101,28,51),(8,112,29,42),(9,103,30,53),(10,114,31,44),(11,105,32,55),(12,116,33,46),(13,107,34,57),(14,118,35,48),(15,109,36,59),(16,120,37,50),(17,111,38,41),(18,102,39,52),(19,113,40,43),(20,104,21,54),(61,92,131,154),(62,83,132,145),(63,94,133,156),(64,85,134,147),(65,96,135,158),(66,87,136,149),(67,98,137,160),(68,89,138,151),(69,100,139,142),(70,91,140,153),(71,82,121,144),(72,93,122,155),(73,84,123,146),(74,95,124,157),(75,86,125,148),(76,97,126,159),(77,88,127,150),(78,99,128,141),(79,90,129,152),(80,81,130,143)], [(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)], [(1,64,32,124),(2,123,33,63),(3,62,34,122),(4,121,35,61),(5,80,36,140),(6,139,37,79),(7,78,38,138),(8,137,39,77),(9,76,40,136),(10,135,21,75),(11,74,22,134),(12,133,23,73),(13,72,24,132),(14,131,25,71),(15,70,26,130),(16,129,27,69),(17,68,28,128),(18,127,29,67),(19,66,30,126),(20,125,31,65),(41,141,101,89),(42,88,102,160),(43,159,103,87),(44,86,104,158),(45,157,105,85),(46,84,106,156),(47,155,107,83),(48,82,108,154),(49,153,109,81),(50,100,110,152),(51,151,111,99),(52,98,112,150),(53,149,113,97),(54,96,114,148),(55,147,115,95),(56,94,116,146),(57,145,117,93),(58,92,118,144),(59,143,119,91),(60,90,120,142)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 32 | 0 | 0 |
| 0 | 0 | 37 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 32 |
| 0 | 0 | 0 | 0 | 37 | 39 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 3 | 0 |
| 0 | 0 | 0 | 22 | 0 | 3 |
| 0 | 0 | 3 | 0 | 19 | 0 |
| 0 | 0 | 0 | 3 | 0 | 19 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 7 |
| 0 | 0 | 0 | 0 | 35 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 40 | 32 | 0 | 0 | 0 | 0 |
| 23 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 30 |
| 0 | 0 | 0 | 0 | 12 | 25 |
| 0 | 0 | 25 | 11 | 0 | 0 |
| 0 | 0 | 29 | 16 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,37,0,0,0,0,32,39,0,0,0,0,0,0,2,37,0,0,0,0,32,39],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,22,0,3,0,0,0,0,22,0,3,0,0,3,0,19,0,0,0,0,3,0,19],[1,0,0,0,0,0,9,40,0,0,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,6,35,0,0,0,0,7,0,0,0],[40,23,0,0,0,0,32,1,0,0,0,0,0,0,0,0,25,29,0,0,0,0,11,16,0,0,16,12,0,0,0,0,30,25,0,0] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4⋊8D10 |
| kernel | C42.100D10 | C20.6Q8 | C4×D20 | C4⋊D20 | D10⋊D4 | D10.13D4 | C20⋊7D4 | C5×C42⋊C2 | C42⋊C2 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{100}D_{10} % in TeX
G:=Group("C4^2.100D10"); // GroupNames label
G:=SmallGroup(320,1207);
// by ID
G=gap.SmallGroup(320,1207);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀