G = C20.76(C4⋊C4) order 320 = 26·5
23rd non-split extension by C20 of C4⋊C4 acting via C4⋊C4/C2×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.76(C4⋊C4), (C2×C20).22Q8, C20.65(C2×Q8), C4⋊C4.234D10, (C2×C20).495D4, C10.84(C4○D8), C10.D8⋊44C2, C20.Q8⋊43C2, (C2×C4).32Dic10, C4.30(C2×Dic10), C42⋊C2.6D5, (C22×C10).73D4, C20.122(C22×C4), (C2×C20).327C23, (C22×C4).337D10, C5⋊4(C23.25D4), C23.37(C5⋊D4), C2.1(D4.8D10), C4.32(C10.D4), C4⋊Dic5.325C22, (C22×C20).148C22, C22.9(C10.D4), C23.21D10.13C2, C4.88(C2×C4×D5), (C2×C5⋊2C8)⋊10C4, C10.63(C2×C4⋊C4), C5⋊2C8.41(C2×C4), (C2×C4).155(C4×D5), (C2×C10).40(C4⋊C4), (C2×C20).262(C2×C4), (C2×C10).456(C2×D4), (C22×C5⋊2C8).7C2, C22.71(C2×C5⋊D4), (C2×C4).273(C5⋊D4), (C5×C4⋊C4).265C22, (C5×C42⋊C2).7C2, C2.14(C2×C10.D4), (C2×C4).427(C22×D5), (C2×C5⋊2C8).250C22, SmallGroup(320,625)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.76(C4⋊C4)
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a9, ac=ca, cbc-1=a10b3 >
Subgroups: 302 in 114 conjugacy classes, 63 normal (25 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, C4⋊C4, C2×C8, C22×C4, Dic5, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C22×C8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×C10, C23.25D4, C2×C5⋊2C8, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C10.D8, C20.Q8, C22×C5⋊2C8, C23.21D10, C5×C42⋊C2, C20.76(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, C4○D8, Dic10, C4×D5, C5⋊D4, C22×D5, C23.25D4, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4, D4.8D10, C20.76(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)
(1 122 145 102 11 132 155 112)(2 131 146 111 12 121 156 101)(3 140 147 120 13 130 157 110)(4 129 148 109 14 139 158 119)(5 138 149 118 15 128 159 108)(6 127 150 107 16 137 160 117)(7 136 151 116 17 126 141 106)(8 125 152 105 18 135 142 115)(9 134 153 114 19 124 143 104)(10 123 154 103 20 133 144 113)(21 60 97 67 31 50 87 77)(22 49 98 76 32 59 88 66)(23 58 99 65 33 48 89 75)(24 47 100 74 34 57 90 64)(25 56 81 63 35 46 91 73)(26 45 82 72 36 55 92 62)(27 54 83 61 37 44 93 71)(28 43 84 70 38 53 94 80)(29 52 85 79 39 42 95 69)(30 41 86 68 40 51 96 78)
(1 27 160 88)(2 28 141 89)(3 29 142 90)(4 30 143 91)(5 31 144 92)(6 32 145 93)(7 33 146 94)(8 34 147 95)(9 35 148 96)(10 36 149 97)(11 37 150 98)(12 38 151 99)(13 39 152 100)(14 40 153 81)(15 21 154 82)(16 22 155 83)(17 23 156 84)(18 24 157 85)(19 25 158 86)(20 26 159 87)(41 124 73 119)(42 125 74 120)(43 126 75 101)(44 127 76 102)(45 128 77 103)(46 129 78 104)(47 130 79 105)(48 131 80 106)(49 132 61 107)(50 133 62 108)(51 134 63 109)(52 135 64 110)(53 136 65 111)(54 137 66 112)(55 138 67 113)(56 139 68 114)(57 140 69 115)(58 121 70 116)(59 122 71 117)(60 123 72 118)
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), (1,122,145,102,11,132,155,112)(2,131,146,111,12,121,156,101)(3,140,147,120,13,130,157,110)(4,129,148,109,14,139,158,119)(5,138,149,118,15,128,159,108)(6,127,150,107,16,137,160,117)(7,136,151,116,17,126,141,106)(8,125,152,105,18,135,142,115)(9,134,153,114,19,124,143,104)(10,123,154,103,20,133,144,113)(21,60,97,67,31,50,87,77)(22,49,98,76,32,59,88,66)(23,58,99,65,33,48,89,75)(24,47,100,74,34,57,90,64)(25,56,81,63,35,46,91,73)(26,45,82,72,36,55,92,62)(27,54,83,61,37,44,93,71)(28,43,84,70,38,53,94,80)(29,52,85,79,39,42,95,69)(30,41,86,68,40,51,96,78), (1,27,160,88)(2,28,141,89)(3,29,142,90)(4,30,143,91)(5,31,144,92)(6,32,145,93)(7,33,146,94)(8,34,147,95)(9,35,148,96)(10,36,149,97)(11,37,150,98)(12,38,151,99)(13,39,152,100)(14,40,153,81)(15,21,154,82)(16,22,155,83)(17,23,156,84)(18,24,157,85)(19,25,158,86)(20,26,159,87)(41,124,73,119)(42,125,74,120)(43,126,75,101)(44,127,76,102)(45,128,77,103)(46,129,78,104)(47,130,79,105)(48,131,80,106)(49,132,61,107)(50,133,62,108)(51,134,63,109)(52,135,64,110)(53,136,65,111)(54,137,66,112)(55,138,67,113)(56,139,68,114)(57,140,69,115)(58,121,70,116)(59,122,71,117)(60,123,72,118)>;
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), (1,122,145,102,11,132,155,112)(2,131,146,111,12,121,156,101)(3,140,147,120,13,130,157,110)(4,129,148,109,14,139,158,119)(5,138,149,118,15,128,159,108)(6,127,150,107,16,137,160,117)(7,136,151,116,17,126,141,106)(8,125,152,105,18,135,142,115)(9,134,153,114,19,124,143,104)(10,123,154,103,20,133,144,113)(21,60,97,67,31,50,87,77)(22,49,98,76,32,59,88,66)(23,58,99,65,33,48,89,75)(24,47,100,74,34,57,90,64)(25,56,81,63,35,46,91,73)(26,45,82,72,36,55,92,62)(27,54,83,61,37,44,93,71)(28,43,84,70,38,53,94,80)(29,52,85,79,39,42,95,69)(30,41,86,68,40,51,96,78), (1,27,160,88)(2,28,141,89)(3,29,142,90)(4,30,143,91)(5,31,144,92)(6,32,145,93)(7,33,146,94)(8,34,147,95)(9,35,148,96)(10,36,149,97)(11,37,150,98)(12,38,151,99)(13,39,152,100)(14,40,153,81)(15,21,154,82)(16,22,155,83)(17,23,156,84)(18,24,157,85)(19,25,158,86)(20,26,159,87)(41,124,73,119)(42,125,74,120)(43,126,75,101)(44,127,76,102)(45,128,77,103)(46,129,78,104)(47,130,79,105)(48,131,80,106)(49,132,61,107)(50,133,62,108)(51,134,63,109)(52,135,64,110)(53,136,65,111)(54,137,66,112)(55,138,67,113)(56,139,68,114)(57,140,69,115)(58,121,70,116)(59,122,71,117)(60,123,72,118) );
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)], [(1,122,145,102,11,132,155,112),(2,131,146,111,12,121,156,101),(3,140,147,120,13,130,157,110),(4,129,148,109,14,139,158,119),(5,138,149,118,15,128,159,108),(6,127,150,107,16,137,160,117),(7,136,151,116,17,126,141,106),(8,125,152,105,18,135,142,115),(9,134,153,114,19,124,143,104),(10,123,154,103,20,133,144,113),(21,60,97,67,31,50,87,77),(22,49,98,76,32,59,88,66),(23,58,99,65,33,48,89,75),(24,47,100,74,34,57,90,64),(25,56,81,63,35,46,91,73),(26,45,82,72,36,55,92,62),(27,54,83,61,37,44,93,71),(28,43,84,70,38,53,94,80),(29,52,85,79,39,42,95,69),(30,41,86,68,40,51,96,78)], [(1,27,160,88),(2,28,141,89),(3,29,142,90),(4,30,143,91),(5,31,144,92),(6,32,145,93),(7,33,146,94),(8,34,147,95),(9,35,148,96),(10,36,149,97),(11,37,150,98),(12,38,151,99),(13,39,152,100),(14,40,153,81),(15,21,154,82),(16,22,155,83),(17,23,156,84),(18,24,157,85),(19,25,158,86),(20,26,159,87),(41,124,73,119),(42,125,74,120),(43,126,75,101),(44,127,76,102),(45,128,77,103),(46,129,78,104),(47,130,79,105),(48,131,80,106),(49,132,61,107),(50,133,62,108),(51,134,63,109),(52,135,64,110),(53,136,65,111),(54,137,66,112),(55,138,67,113),(56,139,68,114),(57,140,69,115),(58,121,70,116),(59,122,71,117),(60,123,72,118)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 8A | ··· | 8H | 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 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D5 | D10 | D10 | C4○D8 | Dic10 | C4×D5 | C5⋊D4 | C5⋊D4 | D4.8D10 |
| kernel | C20.76(C4⋊C4) | C10.D8 | C20.Q8 | C22×C5⋊2C8 | C23.21D10 | C5×C42⋊C2 | C2×C5⋊2C8 | C2×C20 | C2×C20 | C22×C10 | C42⋊C2 | C4⋊C4 | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 8 | 8 | 8 | 4 | 4 | 8 |
Matrix representation of C20.76(C4⋊C4) ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 34 | 8 | 0 | 0 |
| 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 14 | 9 | 0 | 0 |
| 0 | 10 | 27 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 11 | 38 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 17 | 7 | 0 | 0 |
| 0 | 35 | 24 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 28 | 36 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,34,40,0,0,0,8,1,0,0,0,0,0,32,0,0,0,0,0,32],[40,0,0,0,0,0,14,10,0,0,0,9,27,0,0,0,0,0,27,11,0,0,0,0,38],[9,0,0,0,0,0,17,35,0,0,0,7,24,0,0,0,0,0,5,28,0,0,0,5,36] >;
C20.76(C4⋊C4) in GAP, Magma, Sage, TeX
C_{20}._{76}(C_4\rtimes C_4) % in TeX
G:=Group("C20.76(C4:C4)"); // GroupNames label
G:=SmallGroup(320,625);
// by ID
G=gap.SmallGroup(320,625);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^9,a*c=c*a,c*b*c^-1=a^10*b^3>;
// generators/relations