G = C20.47(C4⋊C4) order 320 = 26·5
16th non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C22=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.47(C4⋊C4), (C2×C20).16Q8, C20.63(C2×Q8), C4⋊C4.225D10, (C2×C20).132D4, C4.Dic5⋊13C4, C10.D8⋊24C2, C20.Q8⋊24C2, C4.28(C2×Dic10), (C2×C4).11Dic10, (C22×C4).92D10, C10.82(C8⋊C22), C5⋊4(M4(2)⋊C4), C20.117(C22×C4), (C2×C20).317C23, (C22×C10).182D4, C23.75(C5⋊D4), C2.1(D4.D10), C4.16(C10.D4), C2.1(C20.C23), C10.82(C8.C22), C4⋊Dic5.323C22, (C22×C20).132C22, C23.21D10.12C2, C22.15(C10.D4), C4.86(C2×C4×D5), C5⋊2C8⋊6(C2×C4), (C2×C4⋊C4).6D5, (C10×C4⋊C4).5C2, C10.57(C2×C4⋊C4), (C2×C4).38(C4×D5), (C2×C10).74(C4⋊C4), (C2×C20).249(C2×C4), (C2×C10).437(C2×D4), C2.9(C2×C10.D4), C22.56(C2×C5⋊D4), (C2×C4).181(C5⋊D4), (C5×C4⋊C4).256C22, (C2×C5⋊2C8).79C22, (C2×C4).417(C22×D5), (C2×C4.Dic5).15C2, SmallGroup(320,591)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.47(C4⋊C4)
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, cac-1=a11, cbc-1=a10b3 >
Subgroups: 318 in 118 conjugacy classes, 63 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic5, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×C10, M4(2)⋊C4, C2×C5⋊2C8, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C10.D8, C20.Q8, C2×C4.Dic5, C23.21D10, C10×C4⋊C4, C20.47(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⋊C22, C8.C22, Dic10, C4×D5, C5⋊D4, C22×D5, M4(2)⋊C4, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4, D4.D10, C20.C23, C20.47(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 70 28 139 11 80 38 129)(2 69 29 138 12 79 39 128)(3 68 30 137 13 78 40 127)(4 67 31 136 14 77 21 126)(5 66 32 135 15 76 22 125)(6 65 33 134 16 75 23 124)(7 64 34 133 17 74 24 123)(8 63 35 132 18 73 25 122)(9 62 36 131 19 72 26 121)(10 61 37 130 20 71 27 140)(41 98 153 109 51 88 143 119)(42 97 154 108 52 87 144 118)(43 96 155 107 53 86 145 117)(44 95 156 106 54 85 146 116)(45 94 157 105 55 84 147 115)(46 93 158 104 56 83 148 114)(47 92 159 103 57 82 149 113)(48 91 160 102 58 81 150 112)(49 90 141 101 59 100 151 111)(50 89 142 120 60 99 152 110)
(1 148 23 41)(2 159 24 52)(3 150 25 43)(4 141 26 54)(5 152 27 45)(6 143 28 56)(7 154 29 47)(8 145 30 58)(9 156 31 49)(10 147 32 60)(11 158 33 51)(12 149 34 42)(13 160 35 53)(14 151 36 44)(15 142 37 55)(16 153 38 46)(17 144 39 57)(18 155 40 48)(19 146 21 59)(20 157 22 50)(61 84 135 120)(62 95 136 111)(63 86 137 102)(64 97 138 113)(65 88 139 104)(66 99 140 115)(67 90 121 106)(68 81 122 117)(69 92 123 108)(70 83 124 119)(71 94 125 110)(72 85 126 101)(73 96 127 112)(74 87 128 103)(75 98 129 114)(76 89 130 105)(77 100 131 116)(78 91 132 107)(79 82 133 118)(80 93 134 109)
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), (1,70,28,139,11,80,38,129)(2,69,29,138,12,79,39,128)(3,68,30,137,13,78,40,127)(4,67,31,136,14,77,21,126)(5,66,32,135,15,76,22,125)(6,65,33,134,16,75,23,124)(7,64,34,133,17,74,24,123)(8,63,35,132,18,73,25,122)(9,62,36,131,19,72,26,121)(10,61,37,130,20,71,27,140)(41,98,153,109,51,88,143,119)(42,97,154,108,52,87,144,118)(43,96,155,107,53,86,145,117)(44,95,156,106,54,85,146,116)(45,94,157,105,55,84,147,115)(46,93,158,104,56,83,148,114)(47,92,159,103,57,82,149,113)(48,91,160,102,58,81,150,112)(49,90,141,101,59,100,151,111)(50,89,142,120,60,99,152,110), (1,148,23,41)(2,159,24,52)(3,150,25,43)(4,141,26,54)(5,152,27,45)(6,143,28,56)(7,154,29,47)(8,145,30,58)(9,156,31,49)(10,147,32,60)(11,158,33,51)(12,149,34,42)(13,160,35,53)(14,151,36,44)(15,142,37,55)(16,153,38,46)(17,144,39,57)(18,155,40,48)(19,146,21,59)(20,157,22,50)(61,84,135,120)(62,95,136,111)(63,86,137,102)(64,97,138,113)(65,88,139,104)(66,99,140,115)(67,90,121,106)(68,81,122,117)(69,92,123,108)(70,83,124,119)(71,94,125,110)(72,85,126,101)(73,96,127,112)(74,87,128,103)(75,98,129,114)(76,89,130,105)(77,100,131,116)(78,91,132,107)(79,82,133,118)(80,93,134,109)>;
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), (1,70,28,139,11,80,38,129)(2,69,29,138,12,79,39,128)(3,68,30,137,13,78,40,127)(4,67,31,136,14,77,21,126)(5,66,32,135,15,76,22,125)(6,65,33,134,16,75,23,124)(7,64,34,133,17,74,24,123)(8,63,35,132,18,73,25,122)(9,62,36,131,19,72,26,121)(10,61,37,130,20,71,27,140)(41,98,153,109,51,88,143,119)(42,97,154,108,52,87,144,118)(43,96,155,107,53,86,145,117)(44,95,156,106,54,85,146,116)(45,94,157,105,55,84,147,115)(46,93,158,104,56,83,148,114)(47,92,159,103,57,82,149,113)(48,91,160,102,58,81,150,112)(49,90,141,101,59,100,151,111)(50,89,142,120,60,99,152,110), (1,148,23,41)(2,159,24,52)(3,150,25,43)(4,141,26,54)(5,152,27,45)(6,143,28,56)(7,154,29,47)(8,145,30,58)(9,156,31,49)(10,147,32,60)(11,158,33,51)(12,149,34,42)(13,160,35,53)(14,151,36,44)(15,142,37,55)(16,153,38,46)(17,144,39,57)(18,155,40,48)(19,146,21,59)(20,157,22,50)(61,84,135,120)(62,95,136,111)(63,86,137,102)(64,97,138,113)(65,88,139,104)(66,99,140,115)(67,90,121,106)(68,81,122,117)(69,92,123,108)(70,83,124,119)(71,94,125,110)(72,85,126,101)(73,96,127,112)(74,87,128,103)(75,98,129,114)(76,89,130,105)(77,100,131,116)(78,91,132,107)(79,82,133,118)(80,93,134,109) );
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)], [(1,70,28,139,11,80,38,129),(2,69,29,138,12,79,39,128),(3,68,30,137,13,78,40,127),(4,67,31,136,14,77,21,126),(5,66,32,135,15,76,22,125),(6,65,33,134,16,75,23,124),(7,64,34,133,17,74,24,123),(8,63,35,132,18,73,25,122),(9,62,36,131,19,72,26,121),(10,61,37,130,20,71,27,140),(41,98,153,109,51,88,143,119),(42,97,154,108,52,87,144,118),(43,96,155,107,53,86,145,117),(44,95,156,106,54,85,146,116),(45,94,157,105,55,84,147,115),(46,93,158,104,56,83,148,114),(47,92,159,103,57,82,149,113),(48,91,160,102,58,81,150,112),(49,90,141,101,59,100,151,111),(50,89,142,120,60,99,152,110)], [(1,148,23,41),(2,159,24,52),(3,150,25,43),(4,141,26,54),(5,152,27,45),(6,143,28,56),(7,154,29,47),(8,145,30,58),(9,156,31,49),(10,147,32,60),(11,158,33,51),(12,149,34,42),(13,160,35,53),(14,151,36,44),(15,142,37,55),(16,153,38,46),(17,144,39,57),(18,155,40,48),(19,146,21,59),(20,157,22,50),(61,84,135,120),(62,95,136,111),(63,86,137,102),(64,97,138,113),(65,88,139,104),(66,99,140,115),(67,90,121,106),(68,81,122,117),(69,92,123,108),(70,83,124,119),(71,94,125,110),(72,85,126,101),(73,96,127,112),(74,87,128,103),(75,98,129,114),(76,89,130,105),(77,100,131,116),(78,91,132,107),(79,82,133,118),(80,93,134,109)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D5 | D10 | D10 | Dic10 | C4×D5 | C5⋊D4 | C5⋊D4 | C8⋊C22 | C8.C22 | D4.D10 | C20.C23 |
| kernel | C20.47(C4⋊C4) | C10.D8 | C20.Q8 | C2×C4.Dic5 | C23.21D10 | C10×C4⋊C4 | C4.Dic5 | C2×C20 | C2×C20 | C22×C10 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C20.47(C4⋊C4) ►in GL6(𝔽41)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 16 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 9 | 28 | 0 | 0 | 0 | 0 |
| 3 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 19 | 0 | 0 |
| 0 | 0 | 19 | 38 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 38 |
| 0 | 0 | 0 | 0 | 38 | 22 |
G:=sub<GL(6,GF(41))| [10,16,0,0,0,0,0,37,0,0,0,0,0,0,0,4,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,31,0],[9,3,0,0,0,0,28,32,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,19,0,0,0,0,19,38,0,0,0,0,0,0,19,38,0,0,0,0,38,22] >;
C20.47(C4⋊C4) in GAP, Magma, Sage, TeX
C_{20}._{47}(C_4\rtimes C_4) % in TeX
G:=Group("C20.47(C4:C4)"); // GroupNames label
G:=SmallGroup(320,591);
// by ID
G=gap.SmallGroup(320,591);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b^3>;
// generators/relations