metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.2C8, Dic10.2C8, M4(2).4F5, C5⋊D4.C8, D10.(C2×C8), D5⋊C16⋊5C2, C5⋊1(D4○C16), C20.8(C2×C8), C8.18(C2×F5), C40.18(C2×C4), C4.5(D5⋊C8), C4○D20.2C4, C8⋊D5.2C4, C5⋊C16.2C22, C8.F5⋊4C2, C20.C8⋊3C2, Dic5.2(C2×C8), C4.51(C22×F5), C20.91(C22×C4), C10.11(C22×C8), C5⋊2C8.38C23, C22.1(D5⋊C8), (C8×D5).35C22, (C5×M4(2)).4C4, D20.2C4.3C2, (C2×C5⋊C16)⋊3C2, (C2×C10).1(C2×C8), C2.12(C2×D5⋊C8), (C2×C4).75(C2×F5), (C2×C20).44(C2×C4), C5⋊2C8.20(C2×C4), (C4×D5).45(C2×C4), (C2×C5⋊2C8).188C22, SmallGroup(320,1063)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.C8
G = < a,b,c | a20=1, b2=c8=a10, bab-1=a-1, cac-1=a13, bc=cb >
Subgroups: 226 in 84 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C2×C8, M4(2), M4(2), C4○D4, Dic5, C20, D10, C2×C10, C2×C16, M5(2), C8○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, D4○C16, C5⋊C16, C5⋊C16, C8×D5, C8⋊D5, C2×C5⋊2C8, C5×M4(2), C4○D20, D5⋊C16, C8.F5, C2×C5⋊C16, C20.C8, D20.2C4, Dic10.C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, C22×C8, C2×F5, D4○C16, D5⋊C8, C22×F5, C2×D5⋊C8, Dic10.C8
►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 25 11 35)(2 24 12 34)(3 23 13 33)(4 22 14 32)(5 21 15 31)(6 40 16 30)(7 39 17 29)(8 38 18 28)(9 37 19 27)(10 36 20 26)(41 106 51 116)(42 105 52 115)(43 104 53 114)(44 103 54 113)(45 102 55 112)(46 101 56 111)(47 120 57 110)(48 119 58 109)(49 118 59 108)(50 117 60 107)(61 96 71 86)(62 95 72 85)(63 94 73 84)(64 93 74 83)(65 92 75 82)(66 91 76 81)(67 90 77 100)(68 89 78 99)(69 88 79 98)(70 87 80 97)(121 145 131 155)(122 144 132 154)(123 143 133 153)(124 142 134 152)(125 141 135 151)(126 160 136 150)(127 159 137 149)(128 158 138 148)(129 157 139 147)(130 156 140 146)
(1 156 45 95 40 135 117 67 11 146 55 85 30 125 107 77)(2 153 54 88 21 132 106 80 12 143 44 98 31 122 116 70)(3 150 43 81 22 129 115 73 13 160 53 91 32 139 105 63)(4 147 52 94 23 126 104 66 14 157 42 84 33 136 114 76)(5 144 41 87 24 123 113 79 15 154 51 97 34 133 103 69)(6 141 50 100 25 140 102 72 16 151 60 90 35 130 112 62)(7 158 59 93 26 137 111 65 17 148 49 83 36 127 101 75)(8 155 48 86 27 134 120 78 18 145 58 96 37 124 110 68)(9 152 57 99 28 131 109 71 19 142 47 89 38 121 119 61)(10 149 46 92 29 128 118 64 20 159 56 82 39 138 108 74)
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), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26)(41,106,51,116)(42,105,52,115)(43,104,53,114)(44,103,54,113)(45,102,55,112)(46,101,56,111)(47,120,57,110)(48,119,58,109)(49,118,59,108)(50,117,60,107)(61,96,71,86)(62,95,72,85)(63,94,73,84)(64,93,74,83)(65,92,75,82)(66,91,76,81)(67,90,77,100)(68,89,78,99)(69,88,79,98)(70,87,80,97)(121,145,131,155)(122,144,132,154)(123,143,133,153)(124,142,134,152)(125,141,135,151)(126,160,136,150)(127,159,137,149)(128,158,138,148)(129,157,139,147)(130,156,140,146), (1,156,45,95,40,135,117,67,11,146,55,85,30,125,107,77)(2,153,54,88,21,132,106,80,12,143,44,98,31,122,116,70)(3,150,43,81,22,129,115,73,13,160,53,91,32,139,105,63)(4,147,52,94,23,126,104,66,14,157,42,84,33,136,114,76)(5,144,41,87,24,123,113,79,15,154,51,97,34,133,103,69)(6,141,50,100,25,140,102,72,16,151,60,90,35,130,112,62)(7,158,59,93,26,137,111,65,17,148,49,83,36,127,101,75)(8,155,48,86,27,134,120,78,18,145,58,96,37,124,110,68)(9,152,57,99,28,131,109,71,19,142,47,89,38,121,119,61)(10,149,46,92,29,128,118,64,20,159,56,82,39,138,108,74)>;
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), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26)(41,106,51,116)(42,105,52,115)(43,104,53,114)(44,103,54,113)(45,102,55,112)(46,101,56,111)(47,120,57,110)(48,119,58,109)(49,118,59,108)(50,117,60,107)(61,96,71,86)(62,95,72,85)(63,94,73,84)(64,93,74,83)(65,92,75,82)(66,91,76,81)(67,90,77,100)(68,89,78,99)(69,88,79,98)(70,87,80,97)(121,145,131,155)(122,144,132,154)(123,143,133,153)(124,142,134,152)(125,141,135,151)(126,160,136,150)(127,159,137,149)(128,158,138,148)(129,157,139,147)(130,156,140,146), (1,156,45,95,40,135,117,67,11,146,55,85,30,125,107,77)(2,153,54,88,21,132,106,80,12,143,44,98,31,122,116,70)(3,150,43,81,22,129,115,73,13,160,53,91,32,139,105,63)(4,147,52,94,23,126,104,66,14,157,42,84,33,136,114,76)(5,144,41,87,24,123,113,79,15,154,51,97,34,133,103,69)(6,141,50,100,25,140,102,72,16,151,60,90,35,130,112,62)(7,158,59,93,26,137,111,65,17,148,49,83,36,127,101,75)(8,155,48,86,27,134,120,78,18,145,58,96,37,124,110,68)(9,152,57,99,28,131,109,71,19,142,47,89,38,121,119,61)(10,149,46,92,29,128,118,64,20,159,56,82,39,138,108,74) );
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)], [(1,25,11,35),(2,24,12,34),(3,23,13,33),(4,22,14,32),(5,21,15,31),(6,40,16,30),(7,39,17,29),(8,38,18,28),(9,37,19,27),(10,36,20,26),(41,106,51,116),(42,105,52,115),(43,104,53,114),(44,103,54,113),(45,102,55,112),(46,101,56,111),(47,120,57,110),(48,119,58,109),(49,118,59,108),(50,117,60,107),(61,96,71,86),(62,95,72,85),(63,94,73,84),(64,93,74,83),(65,92,75,82),(66,91,76,81),(67,90,77,100),(68,89,78,99),(69,88,79,98),(70,87,80,97),(121,145,131,155),(122,144,132,154),(123,143,133,153),(124,142,134,152),(125,141,135,151),(126,160,136,150),(127,159,137,149),(128,158,138,148),(129,157,139,147),(130,156,140,146)], [(1,156,45,95,40,135,117,67,11,146,55,85,30,125,107,77),(2,153,54,88,21,132,106,80,12,143,44,98,31,122,116,70),(3,150,43,81,22,129,115,73,13,160,53,91,32,139,105,63),(4,147,52,94,23,126,104,66,14,157,42,84,33,136,114,76),(5,144,41,87,24,123,113,79,15,154,51,97,34,133,103,69),(6,141,50,100,25,140,102,72,16,151,60,90,35,130,112,62),(7,158,59,93,26,137,111,65,17,148,49,83,36,127,101,75),(8,155,48,86,27,134,120,78,18,145,58,96,37,124,110,68),(9,152,57,99,28,131,109,71,19,142,47,89,38,121,119,61),(10,149,46,92,29,128,118,64,20,159,56,82,39,138,108,74)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 16A | ··· | 16H | 16I | ··· | 16T | 20A | 20B | 20C | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 8 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D4○C16 | F5 | C2×F5 | C2×F5 | D5⋊C8 | D5⋊C8 | Dic10.C8 |
| kernel | Dic10.C8 | D5⋊C16 | C8.F5 | C2×C5⋊C16 | C20.C8 | D20.2C4 | C8⋊D5 | C5×M4(2) | C4○D20 | Dic10 | D20 | C5⋊D4 | C5 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of Dic10.C8 ►in GL6(𝔽241)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 0 | 240 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 115 | 0 | 0 | 0 | 0 | 0 |
| 0 | 115 | 0 | 0 | 0 | 0 |
| 0 | 0 | 227 | 14 | 145 | 0 |
| 0 | 0 | 131 | 14 | 0 | 227 |
| 0 | 0 | 227 | 0 | 14 | 131 |
| 0 | 0 | 0 | 145 | 14 | 227 |
G:=sub<GL(6,GF(241))| [0,15,0,0,0,0,16,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,1,1,1],[0,4,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[115,0,0,0,0,0,0,115,0,0,0,0,0,0,227,131,227,0,0,0,14,14,0,145,0,0,145,0,14,14,0,0,0,227,131,227] >;
Dic10.C8 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}.C_8 % in TeX
G:=Group("Dic10.C8"); // GroupNames label
G:=SmallGroup(320,1063);
// by ID
G=gap.SmallGroup(320,1063);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^2=c^8=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^13,b*c=c*b>;
// generators/relations