G = C20.51(C4⋊C4) order 320 = 26·5
20th non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C22=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.51(C4⋊C4), (C2×C20).25Q8, C20.88(C2×Q8), (C2×C20).165D4, (C2×C8).186D10, C20.439(C2×D4), C4⋊Dic5.32C4, C23.27(C4×D5), C10.54(C8○D4), C20.8Q8⋊39C2, C4.53(C2×Dic10), (C2×C4).35Dic10, C23.D5.17C4, (C2×C40).316C22, (C2×C20).864C23, (C22×C4).346D10, (C2×M4(2)).13D5, C4.19(C10.D4), (C10×M4(2)).24C2, C5⋊6(C42.6C22), C2.16(D20.2C4), (C22×C20).178C22, (C4×Dic5).208C22, C23.21D10.15C2, C22.11(C10.D4), C10.72(C2×C4⋊C4), (C2×C4).82(C4×D5), C4.129(C2×C5⋊D4), (C2×C10).43(C4⋊C4), C22.144(C2×C4×D5), (C2×C20).271(C2×C4), (C2×C4).193(C5⋊D4), (C22×C5⋊2C8).10C2, (C2×Dic5).35(C2×C4), C2.17(C2×C10.D4), (C2×C4).806(C22×D5), (C2×C10).235(C22×C4), (C22×C10).132(C2×C4), (C2×C5⋊2C8).330C22, SmallGroup(320,746)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.51(C4⋊C4)
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, cac-1=a11, cbc-1=a10b-1 >
Subgroups: 286 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.6C22, C2×C5⋊2C8, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C22×C20, C20.8Q8, C22×C5⋊2C8, C23.21D10, C10×M4(2), C20.51(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C8○D4, Dic10, C4×D5, C5⋊D4, C22×D5, C42.6C22, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, D20.2C4, C2×C10.D4, C20.51(C4⋊C4)
►On 160 points
Generators in S160
(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 61 158 140)(2 80 159 139)(3 79 160 138)(4 78 141 137)(5 77 142 136)(6 76 143 135)(7 75 144 134)(8 74 145 133)(9 73 146 132)(10 72 147 131)(11 71 148 130)(12 70 149 129)(13 69 150 128)(14 68 151 127)(15 67 152 126)(16 66 153 125)(17 65 154 124)(18 64 155 123)(19 63 156 122)(20 62 157 121)(21 45 93 115)(22 44 94 114)(23 43 95 113)(24 42 96 112)(25 41 97 111)(26 60 98 110)(27 59 99 109)(28 58 100 108)(29 57 81 107)(30 56 82 106)(31 55 83 105)(32 54 84 104)(33 53 85 103)(34 52 86 102)(35 51 87 101)(36 50 88 120)(37 49 89 119)(38 48 90 118)(39 47 91 117)(40 46 92 116)
(1 25 6 40 11 35 16 30)(2 36 7 31 12 26 17 21)(3 27 8 22 13 37 18 32)(4 38 9 33 14 28 19 23)(5 29 10 24 15 39 20 34)(41 125 46 140 51 135 56 130)(42 136 47 131 52 126 57 121)(43 127 48 122 53 137 58 132)(44 138 49 133 54 128 59 123)(45 129 50 124 55 139 60 134)(61 101 76 106 71 111 66 116)(62 112 77 117 72 102 67 107)(63 103 78 108 73 113 68 118)(64 114 79 119 74 104 69 109)(65 105 80 110 75 115 70 120)(81 147 96 152 91 157 86 142)(82 158 97 143 92 148 87 153)(83 149 98 154 93 159 88 144)(84 160 99 145 94 150 89 155)(85 151 100 156 95 141 90 146)
G:=sub<Sym(160)| (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), (1,61,158,140)(2,80,159,139)(3,79,160,138)(4,78,141,137)(5,77,142,136)(6,76,143,135)(7,75,144,134)(8,74,145,133)(9,73,146,132)(10,72,147,131)(11,71,148,130)(12,70,149,129)(13,69,150,128)(14,68,151,127)(15,67,152,126)(16,66,153,125)(17,65,154,124)(18,64,155,123)(19,63,156,122)(20,62,157,121)(21,45,93,115)(22,44,94,114)(23,43,95,113)(24,42,96,112)(25,41,97,111)(26,60,98,110)(27,59,99,109)(28,58,100,108)(29,57,81,107)(30,56,82,106)(31,55,83,105)(32,54,84,104)(33,53,85,103)(34,52,86,102)(35,51,87,101)(36,50,88,120)(37,49,89,119)(38,48,90,118)(39,47,91,117)(40,46,92,116), (1,25,6,40,11,35,16,30)(2,36,7,31,12,26,17,21)(3,27,8,22,13,37,18,32)(4,38,9,33,14,28,19,23)(5,29,10,24,15,39,20,34)(41,125,46,140,51,135,56,130)(42,136,47,131,52,126,57,121)(43,127,48,122,53,137,58,132)(44,138,49,133,54,128,59,123)(45,129,50,124,55,139,60,134)(61,101,76,106,71,111,66,116)(62,112,77,117,72,102,67,107)(63,103,78,108,73,113,68,118)(64,114,79,119,74,104,69,109)(65,105,80,110,75,115,70,120)(81,147,96,152,91,157,86,142)(82,158,97,143,92,148,87,153)(83,149,98,154,93,159,88,144)(84,160,99,145,94,150,89,155)(85,151,100,156,95,141,90,146)>;
G:=Group( (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), (1,61,158,140)(2,80,159,139)(3,79,160,138)(4,78,141,137)(5,77,142,136)(6,76,143,135)(7,75,144,134)(8,74,145,133)(9,73,146,132)(10,72,147,131)(11,71,148,130)(12,70,149,129)(13,69,150,128)(14,68,151,127)(15,67,152,126)(16,66,153,125)(17,65,154,124)(18,64,155,123)(19,63,156,122)(20,62,157,121)(21,45,93,115)(22,44,94,114)(23,43,95,113)(24,42,96,112)(25,41,97,111)(26,60,98,110)(27,59,99,109)(28,58,100,108)(29,57,81,107)(30,56,82,106)(31,55,83,105)(32,54,84,104)(33,53,85,103)(34,52,86,102)(35,51,87,101)(36,50,88,120)(37,49,89,119)(38,48,90,118)(39,47,91,117)(40,46,92,116), (1,25,6,40,11,35,16,30)(2,36,7,31,12,26,17,21)(3,27,8,22,13,37,18,32)(4,38,9,33,14,28,19,23)(5,29,10,24,15,39,20,34)(41,125,46,140,51,135,56,130)(42,136,47,131,52,126,57,121)(43,127,48,122,53,137,58,132)(44,138,49,133,54,128,59,123)(45,129,50,124,55,139,60,134)(61,101,76,106,71,111,66,116)(62,112,77,117,72,102,67,107)(63,103,78,108,73,113,68,118)(64,114,79,119,74,104,69,109)(65,105,80,110,75,115,70,120)(81,147,96,152,91,157,86,142)(82,158,97,143,92,148,87,153)(83,149,98,154,93,159,88,144)(84,160,99,145,94,150,89,155)(85,151,100,156,95,141,90,146) );
G=PermutationGroup([[(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)], [(1,61,158,140),(2,80,159,139),(3,79,160,138),(4,78,141,137),(5,77,142,136),(6,76,143,135),(7,75,144,134),(8,74,145,133),(9,73,146,132),(10,72,147,131),(11,71,148,130),(12,70,149,129),(13,69,150,128),(14,68,151,127),(15,67,152,126),(16,66,153,125),(17,65,154,124),(18,64,155,123),(19,63,156,122),(20,62,157,121),(21,45,93,115),(22,44,94,114),(23,43,95,113),(24,42,96,112),(25,41,97,111),(26,60,98,110),(27,59,99,109),(28,58,100,108),(29,57,81,107),(30,56,82,106),(31,55,83,105),(32,54,84,104),(33,53,85,103),(34,52,86,102),(35,51,87,101),(36,50,88,120),(37,49,89,119),(38,48,90,118),(39,47,91,117),(40,46,92,116)], [(1,25,6,40,11,35,16,30),(2,36,7,31,12,26,17,21),(3,27,8,22,13,37,18,32),(4,38,9,33,14,28,19,23),(5,29,10,24,15,39,20,34),(41,125,46,140,51,135,56,130),(42,136,47,131,52,126,57,121),(43,127,48,122,53,137,58,132),(44,138,49,133,54,128,59,123),(45,129,50,124,55,139,60,134),(61,101,76,106,71,111,66,116),(62,112,77,117,72,102,67,107),(63,103,78,108,73,113,68,118),(64,114,79,119,74,104,69,109),(65,105,80,110,75,115,70,120),(81,147,96,152,91,157,86,142),(82,158,97,143,92,148,87,153),(83,149,98,154,93,159,88,144),(84,160,99,145,94,150,89,155),(85,151,100,156,95,141,90,146)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | D10 | D10 | C8○D4 | Dic10 | C4×D5 | C5⋊D4 | C4×D5 | D20.2C4 |
| kernel | C20.51(C4⋊C4) | C20.8Q8 | C22×C5⋊2C8 | C23.21D10 | C10×M4(2) | C4⋊Dic5 | C23.D5 | C2×C20 | C2×C20 | C2×M4(2) | C2×C8 | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 4 | 8 | 4 | 8 |
Matrix representation of C20.51(C4⋊C4) ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 8 | 32 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 35 | 4 |
| 34 | 26 | 0 | 0 |
| 17 | 7 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 28 | 36 |
| 16 | 5 | 0 | 0 |
| 8 | 25 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 37 | 32 |
G:=sub<GL(4,GF(41))| [9,8,0,0,0,32,0,0,0,0,31,35,0,0,0,4],[34,17,0,0,26,7,0,0,0,0,5,28,0,0,2,36],[16,8,0,0,5,25,0,0,0,0,9,37,0,0,0,32] >;
C20.51(C4⋊C4) in GAP, Magma, Sage, TeX
C_{20}._{51}(C_4\rtimes C_4) % in TeX
G:=Group("C20.51(C4:C4)"); // GroupNames label
G:=SmallGroup(320,746);
// by ID
G=gap.SmallGroup(320,746);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b^-1>;
// generators/relations