metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.93D10, (C4×D20)⋊9C2, C4⋊C4.270D10, (C4×Dic10)⋊9C2, (D5×C42)⋊18C2, D10⋊Q8⋊50C2, D10.7(C4○D4), D10⋊D4.6C2, C42⋊D5⋊30C2, C42⋊C2⋊12D5, (C2×C10).72C24, C22⋊C4.96D10, C4.139(C4○D20), C20.255(C4○D4), (C4×C20).233C22, (C2×C20).147C23, Dic5.8(C4○D4), (C22×C4).193D10, D10.13D4⋊48C2, D10.12D4⋊52C2, C23.84(C22×D5), Dic5.Q8⋊44C2, Dic5.5D4⋊48C2, (C2×D20).215C22, C23.D10⋊48C2, C4⋊Dic5.292C22, (C2×Dic5).25C23, C22.101(C23×D5), C23.D5.95C22, (C22×C20).377C22, (C22×C10).142C23, C5⋊2(C23.36C23), (C4×Dic5).216C22, (C22×D5).175C23, D10⋊C4.143C22, (C2×Dic10).238C22, C10.D4.152C22, (C4×C5⋊D4)⋊52C2, C2.11(D5×C4○D4), C4⋊C4⋊D5⋊49C2, C10.29(C2×C4○D4), C2.31(C2×C4○D20), (C2×C4×D5).373C22, (C5×C42⋊C2)⋊14C2, (C5×C4⋊C4).308C22, (C2×C4).150(C22×D5), (C2×C5⋊D4).110C22, (C5×C22⋊C4).112C22, SmallGroup(320,1200)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 782 in 234 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×3], C10 [×3], C10, C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×5], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×C42, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×2], C22×D5 [×2], C22×C10, C23.36C23, C4×Dic5 [×4], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×4], C2×D20, C2×C5⋊D4 [×2], C22×C20, C4×Dic10, D5×C42, C42⋊D5, C4×D20, C23.D10, D10.12D4, D10⋊D4, Dic5.5D4, Dic5.Q8, D10.13D4, D10⋊Q8, C4⋊C4⋊D5, C4×C5⋊D4 [×2], C5×C42⋊C2, C42.93D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4 [×2], C42.93D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 107 39 68)(2 108 40 69)(3 109 21 70)(4 110 22 71)(5 111 23 72)(6 112 24 73)(7 113 25 74)(8 114 26 75)(9 115 27 76)(10 116 28 77)(11 117 29 78)(12 118 30 79)(13 119 31 80)(14 120 32 61)(15 101 33 62)(16 102 34 63)(17 103 35 64)(18 104 36 65)(19 105 37 66)(20 106 38 67)(41 135 97 145)(42 136 98 146)(43 137 99 147)(44 138 100 148)(45 139 81 149)(46 140 82 150)(47 121 83 151)(48 122 84 152)(49 123 85 153)(50 124 86 154)(51 125 87 155)(52 126 88 156)(53 127 89 157)(54 128 90 158)(55 129 91 159)(56 130 92 160)(57 131 93 141)(58 132 94 142)(59 133 95 143)(60 134 96 144)
(1 132 29 152)(2 143 30 123)(3 134 31 154)(4 145 32 125)(5 136 33 156)(6 147 34 127)(7 138 35 158)(8 149 36 129)(9 140 37 160)(10 151 38 131)(11 122 39 142)(12 153 40 133)(13 124 21 144)(14 155 22 135)(15 126 23 146)(16 157 24 137)(17 128 25 148)(18 159 26 139)(19 130 27 150)(20 141 28 121)(41 61 87 110)(42 101 88 72)(43 63 89 112)(44 103 90 74)(45 65 91 114)(46 105 92 76)(47 67 93 116)(48 107 94 78)(49 69 95 118)(50 109 96 80)(51 71 97 120)(52 111 98 62)(53 73 99 102)(54 113 100 64)(55 75 81 104)(56 115 82 66)(57 77 83 106)(58 117 84 68)(59 79 85 108)(60 119 86 70)
(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 93 11 83)(2 82 12 92)(3 91 13 81)(4 100 14 90)(5 89 15 99)(6 98 16 88)(7 87 17 97)(8 96 18 86)(9 85 19 95)(10 94 20 84)(21 55 31 45)(22 44 32 54)(23 53 33 43)(24 42 34 52)(25 51 35 41)(26 60 36 50)(27 49 37 59)(28 58 38 48)(29 47 39 57)(30 56 40 46)(61 128 71 138)(62 137 72 127)(63 126 73 136)(64 135 74 125)(65 124 75 134)(66 133 76 123)(67 122 77 132)(68 131 78 121)(69 140 79 130)(70 129 80 139)(101 147 111 157)(102 156 112 146)(103 145 113 155)(104 154 114 144)(105 143 115 153)(106 152 116 142)(107 141 117 151)(108 150 118 160)(109 159 119 149)(110 148 120 158)
G:=sub<Sym(160)| (1,107,39,68)(2,108,40,69)(3,109,21,70)(4,110,22,71)(5,111,23,72)(6,112,24,73)(7,113,25,74)(8,114,26,75)(9,115,27,76)(10,116,28,77)(11,117,29,78)(12,118,30,79)(13,119,31,80)(14,120,32,61)(15,101,33,62)(16,102,34,63)(17,103,35,64)(18,104,36,65)(19,105,37,66)(20,106,38,67)(41,135,97,145)(42,136,98,146)(43,137,99,147)(44,138,100,148)(45,139,81,149)(46,140,82,150)(47,121,83,151)(48,122,84,152)(49,123,85,153)(50,124,86,154)(51,125,87,155)(52,126,88,156)(53,127,89,157)(54,128,90,158)(55,129,91,159)(56,130,92,160)(57,131,93,141)(58,132,94,142)(59,133,95,143)(60,134,96,144), (1,132,29,152)(2,143,30,123)(3,134,31,154)(4,145,32,125)(5,136,33,156)(6,147,34,127)(7,138,35,158)(8,149,36,129)(9,140,37,160)(10,151,38,131)(11,122,39,142)(12,153,40,133)(13,124,21,144)(14,155,22,135)(15,126,23,146)(16,157,24,137)(17,128,25,148)(18,159,26,139)(19,130,27,150)(20,141,28,121)(41,61,87,110)(42,101,88,72)(43,63,89,112)(44,103,90,74)(45,65,91,114)(46,105,92,76)(47,67,93,116)(48,107,94,78)(49,69,95,118)(50,109,96,80)(51,71,97,120)(52,111,98,62)(53,73,99,102)(54,113,100,64)(55,75,81,104)(56,115,82,66)(57,77,83,106)(58,117,84,68)(59,79,85,108)(60,119,86,70), (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), (1,93,11,83)(2,82,12,92)(3,91,13,81)(4,100,14,90)(5,89,15,99)(6,98,16,88)(7,87,17,97)(8,96,18,86)(9,85,19,95)(10,94,20,84)(21,55,31,45)(22,44,32,54)(23,53,33,43)(24,42,34,52)(25,51,35,41)(26,60,36,50)(27,49,37,59)(28,58,38,48)(29,47,39,57)(30,56,40,46)(61,128,71,138)(62,137,72,127)(63,126,73,136)(64,135,74,125)(65,124,75,134)(66,133,76,123)(67,122,77,132)(68,131,78,121)(69,140,79,130)(70,129,80,139)(101,147,111,157)(102,156,112,146)(103,145,113,155)(104,154,114,144)(105,143,115,153)(106,152,116,142)(107,141,117,151)(108,150,118,160)(109,159,119,149)(110,148,120,158)>;
G:=Group( (1,107,39,68)(2,108,40,69)(3,109,21,70)(4,110,22,71)(5,111,23,72)(6,112,24,73)(7,113,25,74)(8,114,26,75)(9,115,27,76)(10,116,28,77)(11,117,29,78)(12,118,30,79)(13,119,31,80)(14,120,32,61)(15,101,33,62)(16,102,34,63)(17,103,35,64)(18,104,36,65)(19,105,37,66)(20,106,38,67)(41,135,97,145)(42,136,98,146)(43,137,99,147)(44,138,100,148)(45,139,81,149)(46,140,82,150)(47,121,83,151)(48,122,84,152)(49,123,85,153)(50,124,86,154)(51,125,87,155)(52,126,88,156)(53,127,89,157)(54,128,90,158)(55,129,91,159)(56,130,92,160)(57,131,93,141)(58,132,94,142)(59,133,95,143)(60,134,96,144), (1,132,29,152)(2,143,30,123)(3,134,31,154)(4,145,32,125)(5,136,33,156)(6,147,34,127)(7,138,35,158)(8,149,36,129)(9,140,37,160)(10,151,38,131)(11,122,39,142)(12,153,40,133)(13,124,21,144)(14,155,22,135)(15,126,23,146)(16,157,24,137)(17,128,25,148)(18,159,26,139)(19,130,27,150)(20,141,28,121)(41,61,87,110)(42,101,88,72)(43,63,89,112)(44,103,90,74)(45,65,91,114)(46,105,92,76)(47,67,93,116)(48,107,94,78)(49,69,95,118)(50,109,96,80)(51,71,97,120)(52,111,98,62)(53,73,99,102)(54,113,100,64)(55,75,81,104)(56,115,82,66)(57,77,83,106)(58,117,84,68)(59,79,85,108)(60,119,86,70), (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), (1,93,11,83)(2,82,12,92)(3,91,13,81)(4,100,14,90)(5,89,15,99)(6,98,16,88)(7,87,17,97)(8,96,18,86)(9,85,19,95)(10,94,20,84)(21,55,31,45)(22,44,32,54)(23,53,33,43)(24,42,34,52)(25,51,35,41)(26,60,36,50)(27,49,37,59)(28,58,38,48)(29,47,39,57)(30,56,40,46)(61,128,71,138)(62,137,72,127)(63,126,73,136)(64,135,74,125)(65,124,75,134)(66,133,76,123)(67,122,77,132)(68,131,78,121)(69,140,79,130)(70,129,80,139)(101,147,111,157)(102,156,112,146)(103,145,113,155)(104,154,114,144)(105,143,115,153)(106,152,116,142)(107,141,117,151)(108,150,118,160)(109,159,119,149)(110,148,120,158) );
G=PermutationGroup([(1,107,39,68),(2,108,40,69),(3,109,21,70),(4,110,22,71),(5,111,23,72),(6,112,24,73),(7,113,25,74),(8,114,26,75),(9,115,27,76),(10,116,28,77),(11,117,29,78),(12,118,30,79),(13,119,31,80),(14,120,32,61),(15,101,33,62),(16,102,34,63),(17,103,35,64),(18,104,36,65),(19,105,37,66),(20,106,38,67),(41,135,97,145),(42,136,98,146),(43,137,99,147),(44,138,100,148),(45,139,81,149),(46,140,82,150),(47,121,83,151),(48,122,84,152),(49,123,85,153),(50,124,86,154),(51,125,87,155),(52,126,88,156),(53,127,89,157),(54,128,90,158),(55,129,91,159),(56,130,92,160),(57,131,93,141),(58,132,94,142),(59,133,95,143),(60,134,96,144)], [(1,132,29,152),(2,143,30,123),(3,134,31,154),(4,145,32,125),(5,136,33,156),(6,147,34,127),(7,138,35,158),(8,149,36,129),(9,140,37,160),(10,151,38,131),(11,122,39,142),(12,153,40,133),(13,124,21,144),(14,155,22,135),(15,126,23,146),(16,157,24,137),(17,128,25,148),(18,159,26,139),(19,130,27,150),(20,141,28,121),(41,61,87,110),(42,101,88,72),(43,63,89,112),(44,103,90,74),(45,65,91,114),(46,105,92,76),(47,67,93,116),(48,107,94,78),(49,69,95,118),(50,109,96,80),(51,71,97,120),(52,111,98,62),(53,73,99,102),(54,113,100,64),(55,75,81,104),(56,115,82,66),(57,77,83,106),(58,117,84,68),(59,79,85,108),(60,119,86,70)], [(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)], [(1,93,11,83),(2,82,12,92),(3,91,13,81),(4,100,14,90),(5,89,15,99),(6,98,16,88),(7,87,17,97),(8,96,18,86),(9,85,19,95),(10,94,20,84),(21,55,31,45),(22,44,32,54),(23,53,33,43),(24,42,34,52),(25,51,35,41),(26,60,36,50),(27,49,37,59),(28,58,38,48),(29,47,39,57),(30,56,40,46),(61,128,71,138),(62,137,72,127),(63,126,73,136),(64,135,74,125),(65,124,75,134),(66,133,76,123),(67,122,77,132),(68,131,78,121),(69,140,79,130),(70,129,80,139),(101,147,111,157),(102,156,112,146),(103,145,113,155),(104,154,114,144),(105,143,115,153),(106,152,116,142),(107,141,117,151),(108,150,118,160),(109,159,119,149),(110,148,120,158)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 2 | 0 | 0 |
| 1 | 9 | 0 | 0 |
| 0 | 0 | 11 | 32 |
| 0 | 0 | 9 | 30 |
| 9 | 39 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 19 |
| 0 | 0 | 22 | 19 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 24 | 40 |
| 0 | 0 | 3 | 17 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[32,1,0,0,2,9,0,0,0,0,11,9,0,0,32,30],[9,0,0,0,39,32,0,0,0,0,9,22,0,0,19,19],[9,0,0,0,0,9,0,0,0,0,24,3,0,0,40,17] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 10 | 10 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | D5×C4○D4 |
| kernel | C42.93D10 | C4×Dic10 | D5×C42 | C42⋊D5 | C4×D20 | C23.D10 | D10.12D4 | D10⋊D4 | Dic5.5D4 | Dic5.Q8 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | C4×C5⋊D4 | C5×C42⋊C2 | C42⋊C2 | Dic5 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{93}D_{10} % in TeX
G:=Group("C4^2.93D10"); // GroupNames label
G:=SmallGroup(320,1200);
// by ID
G=gap.SmallGroup(320,1200);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀