G = C20.64(C4⋊C4) order 320 = 26·5
11st non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C2×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.64(C4⋊C4), C20.64(C2×Q8), (C2×C20).21Q8, C4⋊C4.232D10, (C2×C20).471D4, C4.Dic5⋊14C4, C10.D8⋊27C2, C20.Q8⋊27C2, (C2×C4).14Dic10, C4.29(C2×Dic10), C42⋊C2.4D5, (C22×C10).71D4, C2.1(D4⋊D10), C5⋊5(M4(2)⋊C4), (C2×C20).325C23, C20.121(C22×C4), (C22×C4).107D10, C23.53(C5⋊D4), C10.104(C8⋊C22), C2.1(D4.9D10), C4.25(C10.D4), C4⋊Dic5.324C22, C10.104(C8.C22), (C22×C20).145C22, C22.8(C10.D4), C4.87(C2×C4×D5), C5⋊2C8⋊7(C2×C4), C10.62(C2×C4⋊C4), (C2×C4).43(C4×D5), (C2×C10).39(C4⋊C4), (C2×C20).260(C2×C4), (C2×C10).454(C2×D4), (C2×C4⋊Dic5).37C2, C22.70(C2×C5⋊D4), (C2×C4).240(C5⋊D4), (C5×C4⋊C4).263C22, (C5×C42⋊C2).4C2, (C2×C5⋊2C8).85C22, C2.13(C2×C10.D4), (C2×C4).425(C22×D5), (C2×C4.Dic5).16C2, SmallGroup(320,622)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.64(C4⋊C4)
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a-1, ac=ca, cbc-1=a10b3 >
Subgroups: 350 in 118 conjugacy classes, 63 normal (31 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, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C10.D8, C20.Q8, C2×C4.Dic5, C2×C4⋊Dic5, C5×C42⋊C2, C20.64(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, D4.9D10, C20.64(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)
(1 121 33 66 11 131 23 76)(2 140 34 65 12 130 24 75)(3 139 35 64 13 129 25 74)(4 138 36 63 14 128 26 73)(5 137 37 62 15 127 27 72)(6 136 38 61 16 126 28 71)(7 135 39 80 17 125 29 70)(8 134 40 79 18 124 30 69)(9 133 21 78 19 123 31 68)(10 132 22 77 20 122 32 67)(41 157 113 87 51 147 103 97)(42 156 114 86 52 146 104 96)(43 155 115 85 53 145 105 95)(44 154 116 84 54 144 106 94)(45 153 117 83 55 143 107 93)(46 152 118 82 56 142 108 92)(47 151 119 81 57 141 109 91)(48 150 120 100 58 160 110 90)(49 149 101 99 59 159 111 89)(50 148 102 98 60 158 112 88)
(1 115 38 48)(2 116 39 49)(3 117 40 50)(4 118 21 51)(5 119 22 52)(6 120 23 53)(7 101 24 54)(8 102 25 55)(9 103 26 56)(10 104 27 57)(11 105 28 58)(12 106 29 59)(13 107 30 60)(14 108 31 41)(15 109 32 42)(16 110 33 43)(17 111 34 44)(18 112 35 45)(19 113 36 46)(20 114 37 47)(61 90 121 155)(62 91 122 156)(63 92 123 157)(64 93 124 158)(65 94 125 159)(66 95 126 160)(67 96 127 141)(68 97 128 142)(69 98 129 143)(70 99 130 144)(71 100 131 145)(72 81 132 146)(73 82 133 147)(74 83 134 148)(75 84 135 149)(76 85 136 150)(77 86 137 151)(78 87 138 152)(79 88 139 153)(80 89 140 154)
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), (1,121,33,66,11,131,23,76)(2,140,34,65,12,130,24,75)(3,139,35,64,13,129,25,74)(4,138,36,63,14,128,26,73)(5,137,37,62,15,127,27,72)(6,136,38,61,16,126,28,71)(7,135,39,80,17,125,29,70)(8,134,40,79,18,124,30,69)(9,133,21,78,19,123,31,68)(10,132,22,77,20,122,32,67)(41,157,113,87,51,147,103,97)(42,156,114,86,52,146,104,96)(43,155,115,85,53,145,105,95)(44,154,116,84,54,144,106,94)(45,153,117,83,55,143,107,93)(46,152,118,82,56,142,108,92)(47,151,119,81,57,141,109,91)(48,150,120,100,58,160,110,90)(49,149,101,99,59,159,111,89)(50,148,102,98,60,158,112,88), (1,115,38,48)(2,116,39,49)(3,117,40,50)(4,118,21,51)(5,119,22,52)(6,120,23,53)(7,101,24,54)(8,102,25,55)(9,103,26,56)(10,104,27,57)(11,105,28,58)(12,106,29,59)(13,107,30,60)(14,108,31,41)(15,109,32,42)(16,110,33,43)(17,111,34,44)(18,112,35,45)(19,113,36,46)(20,114,37,47)(61,90,121,155)(62,91,122,156)(63,92,123,157)(64,93,124,158)(65,94,125,159)(66,95,126,160)(67,96,127,141)(68,97,128,142)(69,98,129,143)(70,99,130,144)(71,100,131,145)(72,81,132,146)(73,82,133,147)(74,83,134,148)(75,84,135,149)(76,85,136,150)(77,86,137,151)(78,87,138,152)(79,88,139,153)(80,89,140,154)>;
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), (1,121,33,66,11,131,23,76)(2,140,34,65,12,130,24,75)(3,139,35,64,13,129,25,74)(4,138,36,63,14,128,26,73)(5,137,37,62,15,127,27,72)(6,136,38,61,16,126,28,71)(7,135,39,80,17,125,29,70)(8,134,40,79,18,124,30,69)(9,133,21,78,19,123,31,68)(10,132,22,77,20,122,32,67)(41,157,113,87,51,147,103,97)(42,156,114,86,52,146,104,96)(43,155,115,85,53,145,105,95)(44,154,116,84,54,144,106,94)(45,153,117,83,55,143,107,93)(46,152,118,82,56,142,108,92)(47,151,119,81,57,141,109,91)(48,150,120,100,58,160,110,90)(49,149,101,99,59,159,111,89)(50,148,102,98,60,158,112,88), (1,115,38,48)(2,116,39,49)(3,117,40,50)(4,118,21,51)(5,119,22,52)(6,120,23,53)(7,101,24,54)(8,102,25,55)(9,103,26,56)(10,104,27,57)(11,105,28,58)(12,106,29,59)(13,107,30,60)(14,108,31,41)(15,109,32,42)(16,110,33,43)(17,111,34,44)(18,112,35,45)(19,113,36,46)(20,114,37,47)(61,90,121,155)(62,91,122,156)(63,92,123,157)(64,93,124,158)(65,94,125,159)(66,95,126,160)(67,96,127,141)(68,97,128,142)(69,98,129,143)(70,99,130,144)(71,100,131,145)(72,81,132,146)(73,82,133,147)(74,83,134,148)(75,84,135,149)(76,85,136,150)(77,86,137,151)(78,87,138,152)(79,88,139,153)(80,89,140,154) );
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)], [(1,121,33,66,11,131,23,76),(2,140,34,65,12,130,24,75),(3,139,35,64,13,129,25,74),(4,138,36,63,14,128,26,73),(5,137,37,62,15,127,27,72),(6,136,38,61,16,126,28,71),(7,135,39,80,17,125,29,70),(8,134,40,79,18,124,30,69),(9,133,21,78,19,123,31,68),(10,132,22,77,20,122,32,67),(41,157,113,87,51,147,103,97),(42,156,114,86,52,146,104,96),(43,155,115,85,53,145,105,95),(44,154,116,84,54,144,106,94),(45,153,117,83,55,143,107,93),(46,152,118,82,56,142,108,92),(47,151,119,81,57,141,109,91),(48,150,120,100,58,160,110,90),(49,149,101,99,59,159,111,89),(50,148,102,98,60,158,112,88)], [(1,115,38,48),(2,116,39,49),(3,117,40,50),(4,118,21,51),(5,119,22,52),(6,120,23,53),(7,101,24,54),(8,102,25,55),(9,103,26,56),(10,104,27,57),(11,105,28,58),(12,106,29,59),(13,107,30,60),(14,108,31,41),(15,109,32,42),(16,110,33,43),(17,111,34,44),(18,112,35,45),(19,113,36,46),(20,114,37,47),(61,90,121,155),(62,91,122,156),(63,92,123,157),(64,93,124,158),(65,94,125,159),(66,95,126,160),(67,96,127,141),(68,97,128,142),(69,98,129,143),(70,99,130,144),(71,100,131,145),(72,81,132,146),(73,82,133,147),(74,83,134,148),(75,84,135,149),(76,85,136,150),(77,86,137,151),(78,87,138,152),(79,88,139,153),(80,89,140,154)]])
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 | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| 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 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 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 | 4 | 4 | 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 | D4.9D10 |
| kernel | C20.64(C4⋊C4) | C10.D8 | C20.Q8 | C2×C4.Dic5 | C2×C4⋊Dic5 | C5×C42⋊C2 | C4.Dic5 | C2×C20 | C2×C20 | C22×C10 | C42⋊C2 | 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 C20.64(C4⋊C4) ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 11 | 0 | 0 |
| 0 | 0 | 30 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 11 |
| 0 | 0 | 0 | 0 | 30 | 27 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 19 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 17 | 3 | 0 | 0 |
| 0 | 0 | 40 | 24 | 0 | 0 |
| 11 | 32 | 0 | 0 | 0 | 0 |
| 9 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,7,0,0,0,0,0,0,32,30,0,0,0,0,11,27,0,0,0,0,0,0,32,30,0,0,0,0,11,27],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,0,17,40,0,0,0,0,3,24,0,0,9,0,0,0,0,0,19,32,0,0],[11,9,0,0,0,0,32,30,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
C20.64(C4⋊C4) in GAP, Magma, Sage, TeX
C_{20}._{64}(C_4\rtimes C_4) % in TeX
G:=Group("C20.64(C4:C4)"); // GroupNames label
G:=SmallGroup(320,622);
// by ID
G=gap.SmallGroup(320,622);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,387,100,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^10*b^3>;
// generators/relations