metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.2Q8, C10.4SD16, C4.2Dic10, C5⋊2C8⋊2C4, C4⋊C4.2D5, C5⋊2(C4.Q8), C4.12(C4×D5), C20.23(C2×C4), (C2×C10).29D4, (C2×C4).34D10, C2.1(Q8⋊D5), C4⋊Dic5.9C2, C10.10(C4⋊C4), C2.1(D4.D5), (C2×C20).9C22, C2.4(C10.D4), C22.13(C5⋊D4), (C5×C4⋊C4).2C2, (C2×C5⋊2C8).2C2, SmallGroup(160,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.Q8
G = < a,b,c | a20=b4=1, c2=a5b2, bab-1=a11, cac-1=a9, cbc-1=a15b-1 >
Smallest permutation representation of C20.Q8
►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)
(1 142 74 138)(2 153 75 129)(3 144 76 140)(4 155 77 131)(5 146 78 122)(6 157 79 133)(7 148 80 124)(8 159 61 135)(9 150 62 126)(10 141 63 137)(11 152 64 128)(12 143 65 139)(13 154 66 130)(14 145 67 121)(15 156 68 132)(16 147 69 123)(17 158 70 134)(18 149 71 125)(19 160 72 136)(20 151 73 127)(21 91 103 45)(22 82 104 56)(23 93 105 47)(24 84 106 58)(25 95 107 49)(26 86 108 60)(27 97 109 51)(28 88 110 42)(29 99 111 53)(30 90 112 44)(31 81 113 55)(32 92 114 46)(33 83 115 57)(34 94 116 48)(35 85 117 59)(36 96 118 50)(37 87 119 41)(38 98 120 52)(39 89 101 43)(40 100 102 54)
(1 50 79 81 11 60 69 91)(2 59 80 90 12 49 70 100)(3 48 61 99 13 58 71 89)(4 57 62 88 14 47 72 98)(5 46 63 97 15 56 73 87)(6 55 64 86 16 45 74 96)(7 44 65 95 17 54 75 85)(8 53 66 84 18 43 76 94)(9 42 67 93 19 52 77 83)(10 51 68 82 20 41 78 92)(21 157 108 138 31 147 118 128)(22 146 109 127 32 156 119 137)(23 155 110 136 33 145 120 126)(24 144 111 125 34 154 101 135)(25 153 112 134 35 143 102 124)(26 142 113 123 36 152 103 133)(27 151 114 132 37 141 104 122)(28 160 115 121 38 150 105 131)(29 149 116 130 39 159 106 140)(30 158 117 139 40 148 107 129)
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), (1,142,74,138)(2,153,75,129)(3,144,76,140)(4,155,77,131)(5,146,78,122)(6,157,79,133)(7,148,80,124)(8,159,61,135)(9,150,62,126)(10,141,63,137)(11,152,64,128)(12,143,65,139)(13,154,66,130)(14,145,67,121)(15,156,68,132)(16,147,69,123)(17,158,70,134)(18,149,71,125)(19,160,72,136)(20,151,73,127)(21,91,103,45)(22,82,104,56)(23,93,105,47)(24,84,106,58)(25,95,107,49)(26,86,108,60)(27,97,109,51)(28,88,110,42)(29,99,111,53)(30,90,112,44)(31,81,113,55)(32,92,114,46)(33,83,115,57)(34,94,116,48)(35,85,117,59)(36,96,118,50)(37,87,119,41)(38,98,120,52)(39,89,101,43)(40,100,102,54), (1,50,79,81,11,60,69,91)(2,59,80,90,12,49,70,100)(3,48,61,99,13,58,71,89)(4,57,62,88,14,47,72,98)(5,46,63,97,15,56,73,87)(6,55,64,86,16,45,74,96)(7,44,65,95,17,54,75,85)(8,53,66,84,18,43,76,94)(9,42,67,93,19,52,77,83)(10,51,68,82,20,41,78,92)(21,157,108,138,31,147,118,128)(22,146,109,127,32,156,119,137)(23,155,110,136,33,145,120,126)(24,144,111,125,34,154,101,135)(25,153,112,134,35,143,102,124)(26,142,113,123,36,152,103,133)(27,151,114,132,37,141,104,122)(28,160,115,121,38,150,105,131)(29,149,116,130,39,159,106,140)(30,158,117,139,40,148,107,129)>;
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), (1,142,74,138)(2,153,75,129)(3,144,76,140)(4,155,77,131)(5,146,78,122)(6,157,79,133)(7,148,80,124)(8,159,61,135)(9,150,62,126)(10,141,63,137)(11,152,64,128)(12,143,65,139)(13,154,66,130)(14,145,67,121)(15,156,68,132)(16,147,69,123)(17,158,70,134)(18,149,71,125)(19,160,72,136)(20,151,73,127)(21,91,103,45)(22,82,104,56)(23,93,105,47)(24,84,106,58)(25,95,107,49)(26,86,108,60)(27,97,109,51)(28,88,110,42)(29,99,111,53)(30,90,112,44)(31,81,113,55)(32,92,114,46)(33,83,115,57)(34,94,116,48)(35,85,117,59)(36,96,118,50)(37,87,119,41)(38,98,120,52)(39,89,101,43)(40,100,102,54), (1,50,79,81,11,60,69,91)(2,59,80,90,12,49,70,100)(3,48,61,99,13,58,71,89)(4,57,62,88,14,47,72,98)(5,46,63,97,15,56,73,87)(6,55,64,86,16,45,74,96)(7,44,65,95,17,54,75,85)(8,53,66,84,18,43,76,94)(9,42,67,93,19,52,77,83)(10,51,68,82,20,41,78,92)(21,157,108,138,31,147,118,128)(22,146,109,127,32,156,119,137)(23,155,110,136,33,145,120,126)(24,144,111,125,34,154,101,135)(25,153,112,134,35,143,102,124)(26,142,113,123,36,152,103,133)(27,151,114,132,37,141,104,122)(28,160,115,121,38,150,105,131)(29,149,116,130,39,159,106,140)(30,158,117,139,40,148,107,129) );
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)], [(1,142,74,138),(2,153,75,129),(3,144,76,140),(4,155,77,131),(5,146,78,122),(6,157,79,133),(7,148,80,124),(8,159,61,135),(9,150,62,126),(10,141,63,137),(11,152,64,128),(12,143,65,139),(13,154,66,130),(14,145,67,121),(15,156,68,132),(16,147,69,123),(17,158,70,134),(18,149,71,125),(19,160,72,136),(20,151,73,127),(21,91,103,45),(22,82,104,56),(23,93,105,47),(24,84,106,58),(25,95,107,49),(26,86,108,60),(27,97,109,51),(28,88,110,42),(29,99,111,53),(30,90,112,44),(31,81,113,55),(32,92,114,46),(33,83,115,57),(34,94,116,48),(35,85,117,59),(36,96,118,50),(37,87,119,41),(38,98,120,52),(39,89,101,43),(40,100,102,54)], [(1,50,79,81,11,60,69,91),(2,59,80,90,12,49,70,100),(3,48,61,99,13,58,71,89),(4,57,62,88,14,47,72,98),(5,46,63,97,15,56,73,87),(6,55,64,86,16,45,74,96),(7,44,65,95,17,54,75,85),(8,53,66,84,18,43,76,94),(9,42,67,93,19,52,77,83),(10,51,68,82,20,41,78,92),(21,157,108,138,31,147,118,128),(22,146,109,127,32,156,119,137),(23,155,110,136,33,145,120,126),(24,144,111,125,34,154,101,135),(25,153,112,134,35,143,102,124),(26,142,113,123,36,152,103,133),(27,151,114,132,37,141,104,122),(28,160,115,121,38,150,105,131),(29,149,116,130,39,159,106,140),(30,158,117,139,40,148,107,129)]])
C20.Q8 is a maximal subgroup of
Dic5⋊6SD16 D4⋊Dic10 D4.2Dic10 D10.16SD16 D10⋊SD16 C5⋊(C8⋊2D4) C40⋊5C4⋊C2 D4⋊D5⋊6C4 Dic5⋊7SD16 C5⋊Q16⋊5C4 Q8⋊Dic10 Q8.Dic10 D10.11SD16 D10⋊2SD16 (C2×C8).D10 C5⋊2C8.D4 Dic10⋊Q8 C40⋊5Q8 C40⋊3Q8 D5×C4.Q8 C8⋊(C4×D5) D10.12SD16 D10.17SD16 D20⋊Q8 C40⋊4Q8 Dic10.2Q8 C8.6Dic10 C8.27(C4×D5) C40⋊20(C2×C4) C2.D8⋊D5 C2.D8⋊7D5 D20.2Q8 C20.47(C4⋊C4) C4⋊C4.228D10 C4⋊C4.231D10 C20.64(C4⋊C4) C4⋊C4.233D10 C20.76(C4⋊C4) C4⋊C4.236D10 C20.38SD16 D4.3Dic10 C42.48D10 C4×D4.D5 C20.48SD16 Q8.3Dic10 C4×Q8⋊D5 C42.59D10 C4⋊D4.D5 (C2×D4).D10 C4⋊D4⋊D5 C5⋊2C8⋊23D4 C22⋊Q8.D5 C10.(C4○D8) C5⋊2C8⋊24D4 C5⋊(C8.D4) Dic10.4Q8 C42.215D10 C42.68D10 D20.4Q8 C20.SD16 C42.76D10 D20⋊5Q8 Dic10⋊6Q8 C60.7Q8 C30.SD16 C60.2Q8
C20.Q8 is a maximal quotient of
C20.39SD16 C8.Dic10 C40.6Q8 C20.31C42 C60.7Q8 C30.SD16 C60.2Q8
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 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D5 | SD16 | D10 | Dic10 | C4×D5 | C5⋊D4 | D4.D5 | Q8⋊D5 |
| kernel | C20.Q8 | C2×C5⋊2C8 | C4⋊Dic5 | C5×C4⋊C4 | C5⋊2C8 | C20 | C2×C10 | C4⋊C4 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C20.Q8 ►in GL4(𝔽41) generated by
| 7 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 20 |
| 0 | 0 | 4 | 1 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 5 | 32 |
| 21 | 26 | 0 | 0 |
| 2 | 20 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 22 | 11 |
G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,40,4,0,0,20,1],[32,0,0,0,0,32,0,0,0,0,9,5,0,0,0,32],[21,2,0,0,26,20,0,0,0,0,0,22,0,0,28,11] >;
C20.Q8 in GAP, Magma, Sage, TeX
C_{20}.Q_8 % in TeX
G:=Group("C20.Q8"); // GroupNames label
G:=SmallGroup(160,15);
// by ID
G=gap.SmallGroup(160,15);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,313,31,297,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^2=a^5*b^2,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=a^15*b^-1>;
// generators/relations
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