direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C20, C42.3C10, C4⋊C4.6C10, (C4×C20).9C2, C4.4(C2×C20), C2.2(Q8×C10), C20.52(C2×C4), (C2×Q8).5C10, C10.19(C2×Q8), C2.5(C22×C20), (Q8×C10).10C2, C10.40(C4○D4), C10.46(C22×C4), (C2×C10).74C23, (C2×C20).122C22, C22.8(C22×C10), C2.3(C5×C4○D4), (C5×C4⋊C4).13C2, (C2×C4).16(C2×C10), SmallGroup(160,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C20
G = < a,b,c | a20=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 76 in 70 conjugacy classes, 64 normal (16 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, C4×Q8, C2×C20, C2×C20, C5×Q8, C4×C20, C5×C4⋊C4, Q8×C10, Q8×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, Q8, C23, C10, C22×C4, C2×Q8, C4○D4, C20, C2×C10, C4×Q8, C2×C20, C5×Q8, C22×C10, C22×C20, Q8×C10, C5×C4○D4, Q8×C20
►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 59 111 134)(2 60 112 135)(3 41 113 136)(4 42 114 137)(5 43 115 138)(6 44 116 139)(7 45 117 140)(8 46 118 121)(9 47 119 122)(10 48 120 123)(11 49 101 124)(12 50 102 125)(13 51 103 126)(14 52 104 127)(15 53 105 128)(16 54 106 129)(17 55 107 130)(18 56 108 131)(19 57 109 132)(20 58 110 133)(21 83 160 67)(22 84 141 68)(23 85 142 69)(24 86 143 70)(25 87 144 71)(26 88 145 72)(27 89 146 73)(28 90 147 74)(29 91 148 75)(30 92 149 76)(31 93 150 77)(32 94 151 78)(33 95 152 79)(34 96 153 80)(35 97 154 61)(36 98 155 62)(37 99 156 63)(38 100 157 64)(39 81 158 65)(40 82 159 66)
(1 152 111 33)(2 153 112 34)(3 154 113 35)(4 155 114 36)(5 156 115 37)(6 157 116 38)(7 158 117 39)(8 159 118 40)(9 160 119 21)(10 141 120 22)(11 142 101 23)(12 143 102 24)(13 144 103 25)(14 145 104 26)(15 146 105 27)(16 147 106 28)(17 148 107 29)(18 149 108 30)(19 150 109 31)(20 151 110 32)(41 97 136 61)(42 98 137 62)(43 99 138 63)(44 100 139 64)(45 81 140 65)(46 82 121 66)(47 83 122 67)(48 84 123 68)(49 85 124 69)(50 86 125 70)(51 87 126 71)(52 88 127 72)(53 89 128 73)(54 90 129 74)(55 91 130 75)(56 92 131 76)(57 93 132 77)(58 94 133 78)(59 95 134 79)(60 96 135 80)
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), (1,59,111,134)(2,60,112,135)(3,41,113,136)(4,42,114,137)(5,43,115,138)(6,44,116,139)(7,45,117,140)(8,46,118,121)(9,47,119,122)(10,48,120,123)(11,49,101,124)(12,50,102,125)(13,51,103,126)(14,52,104,127)(15,53,105,128)(16,54,106,129)(17,55,107,130)(18,56,108,131)(19,57,109,132)(20,58,110,133)(21,83,160,67)(22,84,141,68)(23,85,142,69)(24,86,143,70)(25,87,144,71)(26,88,145,72)(27,89,146,73)(28,90,147,74)(29,91,148,75)(30,92,149,76)(31,93,150,77)(32,94,151,78)(33,95,152,79)(34,96,153,80)(35,97,154,61)(36,98,155,62)(37,99,156,63)(38,100,157,64)(39,81,158,65)(40,82,159,66), (1,152,111,33)(2,153,112,34)(3,154,113,35)(4,155,114,36)(5,156,115,37)(6,157,116,38)(7,158,117,39)(8,159,118,40)(9,160,119,21)(10,141,120,22)(11,142,101,23)(12,143,102,24)(13,144,103,25)(14,145,104,26)(15,146,105,27)(16,147,106,28)(17,148,107,29)(18,149,108,30)(19,150,109,31)(20,151,110,32)(41,97,136,61)(42,98,137,62)(43,99,138,63)(44,100,139,64)(45,81,140,65)(46,82,121,66)(47,83,122,67)(48,84,123,68)(49,85,124,69)(50,86,125,70)(51,87,126,71)(52,88,127,72)(53,89,128,73)(54,90,129,74)(55,91,130,75)(56,92,131,76)(57,93,132,77)(58,94,133,78)(59,95,134,79)(60,96,135,80)>;
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), (1,59,111,134)(2,60,112,135)(3,41,113,136)(4,42,114,137)(5,43,115,138)(6,44,116,139)(7,45,117,140)(8,46,118,121)(9,47,119,122)(10,48,120,123)(11,49,101,124)(12,50,102,125)(13,51,103,126)(14,52,104,127)(15,53,105,128)(16,54,106,129)(17,55,107,130)(18,56,108,131)(19,57,109,132)(20,58,110,133)(21,83,160,67)(22,84,141,68)(23,85,142,69)(24,86,143,70)(25,87,144,71)(26,88,145,72)(27,89,146,73)(28,90,147,74)(29,91,148,75)(30,92,149,76)(31,93,150,77)(32,94,151,78)(33,95,152,79)(34,96,153,80)(35,97,154,61)(36,98,155,62)(37,99,156,63)(38,100,157,64)(39,81,158,65)(40,82,159,66), (1,152,111,33)(2,153,112,34)(3,154,113,35)(4,155,114,36)(5,156,115,37)(6,157,116,38)(7,158,117,39)(8,159,118,40)(9,160,119,21)(10,141,120,22)(11,142,101,23)(12,143,102,24)(13,144,103,25)(14,145,104,26)(15,146,105,27)(16,147,106,28)(17,148,107,29)(18,149,108,30)(19,150,109,31)(20,151,110,32)(41,97,136,61)(42,98,137,62)(43,99,138,63)(44,100,139,64)(45,81,140,65)(46,82,121,66)(47,83,122,67)(48,84,123,68)(49,85,124,69)(50,86,125,70)(51,87,126,71)(52,88,127,72)(53,89,128,73)(54,90,129,74)(55,91,130,75)(56,92,131,76)(57,93,132,77)(58,94,133,78)(59,95,134,79)(60,96,135,80) );
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)], [(1,59,111,134),(2,60,112,135),(3,41,113,136),(4,42,114,137),(5,43,115,138),(6,44,116,139),(7,45,117,140),(8,46,118,121),(9,47,119,122),(10,48,120,123),(11,49,101,124),(12,50,102,125),(13,51,103,126),(14,52,104,127),(15,53,105,128),(16,54,106,129),(17,55,107,130),(18,56,108,131),(19,57,109,132),(20,58,110,133),(21,83,160,67),(22,84,141,68),(23,85,142,69),(24,86,143,70),(25,87,144,71),(26,88,145,72),(27,89,146,73),(28,90,147,74),(29,91,148,75),(30,92,149,76),(31,93,150,77),(32,94,151,78),(33,95,152,79),(34,96,153,80),(35,97,154,61),(36,98,155,62),(37,99,156,63),(38,100,157,64),(39,81,158,65),(40,82,159,66)], [(1,152,111,33),(2,153,112,34),(3,154,113,35),(4,155,114,36),(5,156,115,37),(6,157,116,38),(7,158,117,39),(8,159,118,40),(9,160,119,21),(10,141,120,22),(11,142,101,23),(12,143,102,24),(13,144,103,25),(14,145,104,26),(15,146,105,27),(16,147,106,28),(17,148,107,29),(18,149,108,30),(19,150,109,31),(20,151,110,32),(41,97,136,61),(42,98,137,62),(43,99,138,63),(44,100,139,64),(45,81,140,65),(46,82,121,66),(47,83,122,67),(48,84,123,68),(49,85,124,69),(50,86,125,70),(51,87,126,71),(52,88,127,72),(53,89,128,73),(54,90,129,74),(55,91,130,75),(56,92,131,76),(57,93,132,77),(58,94,133,78),(59,95,134,79),(60,96,135,80)]])
Q8×C20 is a maximal subgroup of
C20.26Q16 C20.48SD16 C20.23Q16 Q8.3Dic10 C42.210D10 C42.56D10 Q8⋊D20 Q8.1D20 C42.59D10 C20⋊7Q16 Dic10⋊10Q8 C42.122D10 Q8⋊5Dic10 Q8⋊6Dic10 C42.125D10 C42.126D10 Q8⋊5D20 Q8⋊6D20 C42.232D10 D20⋊10Q8 C42.131D10 C42.132D10 C42.133D10 C42.134D10 C42.135D10 C42.136D10
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | ··· | 20BL |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | Q8 | C4○D4 | C5×Q8 | C5×C4○D4 |
| kernel | Q8×C20 | C4×C20 | C5×C4⋊C4 | Q8×C10 | C5×Q8 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C20 | C10 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 8 | 4 | 12 | 12 | 4 | 32 | 2 | 2 | 8 | 8 |
Matrix representation of Q8×C20 ►in GL3(𝔽41) generated by
| 32 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 40 | 0 | 0 |
| 0 | 12 | 40 |
| 0 | 22 | 29 |
| 40 | 0 | 0 |
| 0 | 16 | 34 |
| 0 | 25 | 25 |
G:=sub<GL(3,GF(41))| [32,0,0,0,2,0,0,0,2],[40,0,0,0,12,22,0,40,29],[40,0,0,0,16,25,0,34,25] >;
Q8×C20 in GAP, Magma, Sage, TeX
Q_8\times C_{20} % in TeX
G:=Group("Q8xC20"); // GroupNames label
G:=SmallGroup(160,180);
// by ID
G=gap.SmallGroup(160,180);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,247,554]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations