metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.6D8, C20.1Q8, C10.3Q16, C4.1Dic10, C5⋊2C8⋊1C4, C4⋊C4.1D5, C5⋊2(C2.D8), C4.11(C4×D5), C10.9(C4⋊C4), C20.22(C2×C4), C2.1(D4⋊D5), (C2×C10).28D4, (C2×C4).33D10, C4⋊Dic5.8C2, (C2×C20).8C22, C2.1(C5⋊Q16), C2.3(C10.D4), C22.12(C5⋊D4), (C5×C4⋊C4).1C2, (C2×C5⋊2C8).1C2, SmallGroup(160,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.D8
G = < a,b,c | a10=b8=1, c2=a5, bab-1=cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C10.D8
►Regular action 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)
(1 125 50 117 33 108 12 91)(2 124 41 116 34 107 13 100)(3 123 42 115 35 106 14 99)(4 122 43 114 36 105 15 98)(5 121 44 113 37 104 16 97)(6 130 45 112 38 103 17 96)(7 129 46 111 39 102 18 95)(8 128 47 120 40 101 19 94)(9 127 48 119 31 110 20 93)(10 126 49 118 32 109 11 92)(21 90 132 53 149 61 158 72)(22 89 133 52 150 70 159 71)(23 88 134 51 141 69 160 80)(24 87 135 60 142 68 151 79)(25 86 136 59 143 67 152 78)(26 85 137 58 144 66 153 77)(27 84 138 57 145 65 154 76)(28 83 139 56 146 64 155 75)(29 82 140 55 147 63 156 74)(30 81 131 54 148 62 157 73)
(1 136 6 131)(2 135 7 140)(3 134 8 139)(4 133 9 138)(5 132 10 137)(11 144 16 149)(12 143 17 148)(13 142 18 147)(14 141 19 146)(15 150 20 145)(21 49 26 44)(22 48 27 43)(23 47 28 42)(24 46 29 41)(25 45 30 50)(31 154 36 159)(32 153 37 158)(33 152 38 157)(34 151 39 156)(35 160 40 155)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 119 76 114)(72 118 77 113)(73 117 78 112)(74 116 79 111)(75 115 80 120)(81 125 86 130)(82 124 87 129)(83 123 88 128)(84 122 89 127)(85 121 90 126)
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), (1,125,50,117,33,108,12,91)(2,124,41,116,34,107,13,100)(3,123,42,115,35,106,14,99)(4,122,43,114,36,105,15,98)(5,121,44,113,37,104,16,97)(6,130,45,112,38,103,17,96)(7,129,46,111,39,102,18,95)(8,128,47,120,40,101,19,94)(9,127,48,119,31,110,20,93)(10,126,49,118,32,109,11,92)(21,90,132,53,149,61,158,72)(22,89,133,52,150,70,159,71)(23,88,134,51,141,69,160,80)(24,87,135,60,142,68,151,79)(25,86,136,59,143,67,152,78)(26,85,137,58,144,66,153,77)(27,84,138,57,145,65,154,76)(28,83,139,56,146,64,155,75)(29,82,140,55,147,63,156,74)(30,81,131,54,148,62,157,73), (1,136,6,131)(2,135,7,140)(3,134,8,139)(4,133,9,138)(5,132,10,137)(11,144,16,149)(12,143,17,148)(13,142,18,147)(14,141,19,146)(15,150,20,145)(21,49,26,44)(22,48,27,43)(23,47,28,42)(24,46,29,41)(25,45,30,50)(31,154,36,159)(32,153,37,158)(33,152,38,157)(34,151,39,156)(35,160,40,155)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,119,76,114)(72,118,77,113)(73,117,78,112)(74,116,79,111)(75,115,80,120)(81,125,86,130)(82,124,87,129)(83,123,88,128)(84,122,89,127)(85,121,90,126)>;
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), (1,125,50,117,33,108,12,91)(2,124,41,116,34,107,13,100)(3,123,42,115,35,106,14,99)(4,122,43,114,36,105,15,98)(5,121,44,113,37,104,16,97)(6,130,45,112,38,103,17,96)(7,129,46,111,39,102,18,95)(8,128,47,120,40,101,19,94)(9,127,48,119,31,110,20,93)(10,126,49,118,32,109,11,92)(21,90,132,53,149,61,158,72)(22,89,133,52,150,70,159,71)(23,88,134,51,141,69,160,80)(24,87,135,60,142,68,151,79)(25,86,136,59,143,67,152,78)(26,85,137,58,144,66,153,77)(27,84,138,57,145,65,154,76)(28,83,139,56,146,64,155,75)(29,82,140,55,147,63,156,74)(30,81,131,54,148,62,157,73), (1,136,6,131)(2,135,7,140)(3,134,8,139)(4,133,9,138)(5,132,10,137)(11,144,16,149)(12,143,17,148)(13,142,18,147)(14,141,19,146)(15,150,20,145)(21,49,26,44)(22,48,27,43)(23,47,28,42)(24,46,29,41)(25,45,30,50)(31,154,36,159)(32,153,37,158)(33,152,38,157)(34,151,39,156)(35,160,40,155)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,119,76,114)(72,118,77,113)(73,117,78,112)(74,116,79,111)(75,115,80,120)(81,125,86,130)(82,124,87,129)(83,123,88,128)(84,122,89,127)(85,121,90,126) );
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)], [(1,125,50,117,33,108,12,91),(2,124,41,116,34,107,13,100),(3,123,42,115,35,106,14,99),(4,122,43,114,36,105,15,98),(5,121,44,113,37,104,16,97),(6,130,45,112,38,103,17,96),(7,129,46,111,39,102,18,95),(8,128,47,120,40,101,19,94),(9,127,48,119,31,110,20,93),(10,126,49,118,32,109,11,92),(21,90,132,53,149,61,158,72),(22,89,133,52,150,70,159,71),(23,88,134,51,141,69,160,80),(24,87,135,60,142,68,151,79),(25,86,136,59,143,67,152,78),(26,85,137,58,144,66,153,77),(27,84,138,57,145,65,154,76),(28,83,139,56,146,64,155,75),(29,82,140,55,147,63,156,74),(30,81,131,54,148,62,157,73)], [(1,136,6,131),(2,135,7,140),(3,134,8,139),(4,133,9,138),(5,132,10,137),(11,144,16,149),(12,143,17,148),(13,142,18,147),(14,141,19,146),(15,150,20,145),(21,49,26,44),(22,48,27,43),(23,47,28,42),(24,46,29,41),(25,45,30,50),(31,154,36,159),(32,153,37,158),(33,152,38,157),(34,151,39,156),(35,160,40,155),(51,94,56,99),(52,93,57,98),(53,92,58,97),(54,91,59,96),(55,100,60,95),(61,109,66,104),(62,108,67,103),(63,107,68,102),(64,106,69,101),(65,105,70,110),(71,119,76,114),(72,118,77,113),(73,117,78,112),(74,116,79,111),(75,115,80,120),(81,125,86,130),(82,124,87,129),(83,123,88,128),(84,122,89,127),(85,121,90,126)]])
C10.D8 is a maximal subgroup of
Dic5⋊4D8 D4.D5⋊5C4 Dic5.14D8 D4.Dic10 D10.12D8 D10⋊D8 C40⋊6C4⋊C2 C5⋊2C8⋊D4 Dic5⋊4Q16 Dic5.9Q16 Q8.2Dic10 D10.7Q16 C5⋊(C8⋊D4) D10⋊Q16 D10⋊1C8.C2 Q8⋊D5⋊6C4 C40⋊3Q8 Dic10.Q8 C8.8Dic10 (C8×D5)⋊C4 C8⋊(C4×D5) C4.Q8⋊D5 C20.(C4○D4) D20.Q8 C40⋊2Q8 Dic10⋊2Q8 C40⋊4Q8 D5×C2.D8 C40⋊20(C2×C4) D10.13D8 D10.8Q16 D20⋊2Q8 C20.47(C4⋊C4) (C2×C10).40D8 C4⋊C4.230D10 C20.64(C4⋊C4) C4⋊C4.233D10 C20.76(C4⋊C4) C4⋊C4.236D10 C20.50D8 D4.3Dic10 C4×D4⋊D5 C42.51D10 C20.23Q16 Q8.3Dic10 C42.56D10 C4×C5⋊Q16 (C2×C10).D8 (C2×D4).D10 (C2×C10)⋊D8 C4.(D4×D5) (C2×C10).Q16 C10.(C4○D8) C22⋊Q8⋊D5 (C2×C10)⋊Q16 Dic10.4Q8 C42.215D10 C42.68D10 D20.4Q8 C20.17D8 C42.76D10 D20⋊6Q8 Dic10⋊5Q8 C60.8Q8 C30.20D8 C60.1Q8
C10.D8 is a maximal quotient of
C20.53D8 C40.2Q8 C10.SD32 C40.7Q8 C20.31C42 C60.8Q8 C30.20D8 C60.1Q8
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | D8 | Q16 | D10 | Dic10 | C4×D5 | C5⋊D4 | D4⋊D5 | C5⋊Q16 |
| kernel | C10.D8 | C2×C5⋊2C8 | C4⋊Dic5 | C5×C4⋊C4 | C5⋊2C8 | C20 | C2×C10 | C4⋊C4 | C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C10.D8 ►in GL4(𝔽41) generated by
| 0 | 6 | 0 | 0 |
| 34 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 24 | 35 | 0 | 0 |
| 21 | 17 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 15 | 17 |
| 9 | 0 | 0 | 0 |
| 31 | 32 | 0 | 0 |
| 0 | 0 | 33 | 26 |
| 0 | 0 | 18 | 8 |
G:=sub<GL(4,GF(41))| [0,34,0,0,6,7,0,0,0,0,40,0,0,0,0,40],[24,21,0,0,35,17,0,0,0,0,0,15,0,0,30,17],[9,31,0,0,0,32,0,0,0,0,33,18,0,0,26,8] >;
C10.D8 in GAP, Magma, Sage, TeX
C_{10}.D_8 % in TeX
G:=Group("C10.D8"); // GroupNames label
G:=SmallGroup(160,14);
// by ID
G=gap.SmallGroup(160,14);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,121,31,297,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^10=b^8=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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