metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.8Q8, Dic5⋊1C8, C20.51D4, C4.8Dic10, C10.6M4(2), C5⋊4(C4⋊C8), (C2×C8).1D5, C2.4(C8×D5), (C2×C40).1C2, C10.13(C2×C8), (C2×C4).91D10, C10.11(C4⋊C4), C22.9(C4×D5), C2.1(C8⋊D5), C4.26(C5⋊D4), (C4×Dic5).5C2, (C2×Dic5).3C4, (C2×C20).105C22, C2.1(C10.D4), (C2×C5⋊2C8).9C2, (C2×C10).30(C2×C4), SmallGroup(160,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.8Q8
G = < a,b,c | a20=1, b4=a10, c2=a15b2, ab=ba, cac-1=a9, cbc-1=a15b3 >
Smallest permutation representation of C20.8Q8
►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 81 130 107 11 91 140 117)(2 82 131 108 12 92 121 118)(3 83 132 109 13 93 122 119)(4 84 133 110 14 94 123 120)(5 85 134 111 15 95 124 101)(6 86 135 112 16 96 125 102)(7 87 136 113 17 97 126 103)(8 88 137 114 18 98 127 104)(9 89 138 115 19 99 128 105)(10 90 139 116 20 100 129 106)(21 145 79 59 31 155 69 49)(22 146 80 60 32 156 70 50)(23 147 61 41 33 157 71 51)(24 148 62 42 34 158 72 52)(25 149 63 43 35 159 73 53)(26 150 64 44 36 160 74 54)(27 151 65 45 37 141 75 55)(28 152 66 46 38 142 76 56)(29 153 67 47 39 143 77 57)(30 154 68 48 40 144 78 58)
(1 80 125 27)(2 69 126 36)(3 78 127 25)(4 67 128 34)(5 76 129 23)(6 65 130 32)(7 74 131 21)(8 63 132 30)(9 72 133 39)(10 61 134 28)(11 70 135 37)(12 79 136 26)(13 68 137 35)(14 77 138 24)(15 66 139 33)(16 75 140 22)(17 64 121 31)(18 73 122 40)(19 62 123 29)(20 71 124 38)(41 90 152 111)(42 99 153 120)(43 88 154 109)(44 97 155 118)(45 86 156 107)(46 95 157 116)(47 84 158 105)(48 93 159 114)(49 82 160 103)(50 91 141 112)(51 100 142 101)(52 89 143 110)(53 98 144 119)(54 87 145 108)(55 96 146 117)(56 85 147 106)(57 94 148 115)(58 83 149 104)(59 92 150 113)(60 81 151 102)
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), (1,81,130,107,11,91,140,117)(2,82,131,108,12,92,121,118)(3,83,132,109,13,93,122,119)(4,84,133,110,14,94,123,120)(5,85,134,111,15,95,124,101)(6,86,135,112,16,96,125,102)(7,87,136,113,17,97,126,103)(8,88,137,114,18,98,127,104)(9,89,138,115,19,99,128,105)(10,90,139,116,20,100,129,106)(21,145,79,59,31,155,69,49)(22,146,80,60,32,156,70,50)(23,147,61,41,33,157,71,51)(24,148,62,42,34,158,72,52)(25,149,63,43,35,159,73,53)(26,150,64,44,36,160,74,54)(27,151,65,45,37,141,75,55)(28,152,66,46,38,142,76,56)(29,153,67,47,39,143,77,57)(30,154,68,48,40,144,78,58), (1,80,125,27)(2,69,126,36)(3,78,127,25)(4,67,128,34)(5,76,129,23)(6,65,130,32)(7,74,131,21)(8,63,132,30)(9,72,133,39)(10,61,134,28)(11,70,135,37)(12,79,136,26)(13,68,137,35)(14,77,138,24)(15,66,139,33)(16,75,140,22)(17,64,121,31)(18,73,122,40)(19,62,123,29)(20,71,124,38)(41,90,152,111)(42,99,153,120)(43,88,154,109)(44,97,155,118)(45,86,156,107)(46,95,157,116)(47,84,158,105)(48,93,159,114)(49,82,160,103)(50,91,141,112)(51,100,142,101)(52,89,143,110)(53,98,144,119)(54,87,145,108)(55,96,146,117)(56,85,147,106)(57,94,148,115)(58,83,149,104)(59,92,150,113)(60,81,151,102)>;
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), (1,81,130,107,11,91,140,117)(2,82,131,108,12,92,121,118)(3,83,132,109,13,93,122,119)(4,84,133,110,14,94,123,120)(5,85,134,111,15,95,124,101)(6,86,135,112,16,96,125,102)(7,87,136,113,17,97,126,103)(8,88,137,114,18,98,127,104)(9,89,138,115,19,99,128,105)(10,90,139,116,20,100,129,106)(21,145,79,59,31,155,69,49)(22,146,80,60,32,156,70,50)(23,147,61,41,33,157,71,51)(24,148,62,42,34,158,72,52)(25,149,63,43,35,159,73,53)(26,150,64,44,36,160,74,54)(27,151,65,45,37,141,75,55)(28,152,66,46,38,142,76,56)(29,153,67,47,39,143,77,57)(30,154,68,48,40,144,78,58), (1,80,125,27)(2,69,126,36)(3,78,127,25)(4,67,128,34)(5,76,129,23)(6,65,130,32)(7,74,131,21)(8,63,132,30)(9,72,133,39)(10,61,134,28)(11,70,135,37)(12,79,136,26)(13,68,137,35)(14,77,138,24)(15,66,139,33)(16,75,140,22)(17,64,121,31)(18,73,122,40)(19,62,123,29)(20,71,124,38)(41,90,152,111)(42,99,153,120)(43,88,154,109)(44,97,155,118)(45,86,156,107)(46,95,157,116)(47,84,158,105)(48,93,159,114)(49,82,160,103)(50,91,141,112)(51,100,142,101)(52,89,143,110)(53,98,144,119)(54,87,145,108)(55,96,146,117)(56,85,147,106)(57,94,148,115)(58,83,149,104)(59,92,150,113)(60,81,151,102) );
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)], [(1,81,130,107,11,91,140,117),(2,82,131,108,12,92,121,118),(3,83,132,109,13,93,122,119),(4,84,133,110,14,94,123,120),(5,85,134,111,15,95,124,101),(6,86,135,112,16,96,125,102),(7,87,136,113,17,97,126,103),(8,88,137,114,18,98,127,104),(9,89,138,115,19,99,128,105),(10,90,139,116,20,100,129,106),(21,145,79,59,31,155,69,49),(22,146,80,60,32,156,70,50),(23,147,61,41,33,157,71,51),(24,148,62,42,34,158,72,52),(25,149,63,43,35,159,73,53),(26,150,64,44,36,160,74,54),(27,151,65,45,37,141,75,55),(28,152,66,46,38,142,76,56),(29,153,67,47,39,143,77,57),(30,154,68,48,40,144,78,58)], [(1,80,125,27),(2,69,126,36),(3,78,127,25),(4,67,128,34),(5,76,129,23),(6,65,130,32),(7,74,131,21),(8,63,132,30),(9,72,133,39),(10,61,134,28),(11,70,135,37),(12,79,136,26),(13,68,137,35),(14,77,138,24),(15,66,139,33),(16,75,140,22),(17,64,121,31),(18,73,122,40),(19,62,123,29),(20,71,124,38),(41,90,152,111),(42,99,153,120),(43,88,154,109),(44,97,155,118),(45,86,156,107),(46,95,157,116),(47,84,158,105),(48,93,159,114),(49,82,160,103),(50,91,141,112),(51,100,142,101),(52,89,143,110),(53,98,144,119),(54,87,145,108),(55,96,146,117),(56,85,147,106),(57,94,148,115),(58,83,149,104),(59,92,150,113),(60,81,151,102)]])
C20.8Q8 is a maximal subgroup of
C8×Dic10 C40⋊11Q8 C42.282D10 C42.243D10 C40⋊Q8 C42.182D10 C42.185D10 Dic5.14M4(2) Dic5.9M4(2) C40⋊8C4⋊C2 C5⋊5(C8×D4) D10⋊4M4(2) Dic5⋊2M4(2) C5⋊2C8⋊26D4 Dic5.14D8 D4⋊Dic10 Dic10⋊2D4 D4.Dic10 D4.2Dic10 Dic10.D4 D20⋊3D4 D20.D4 Q8⋊Dic10 Dic5⋊Q16 Dic5.9Q16 Q8.Dic10 Dic10.11D4 Q8.2Dic10 Dic5⋊SD16 D20.12D4 Dic5.5M4(2) Dic10⋊5C8 C42.198D10 D5×C4⋊C8 C20⋊5M4(2) C42.30D10 Dic10⋊Q8 Dic10.Q8 D20⋊Q8 D20.Q8 Dic10⋊2Q8 Dic10.2Q8 D20⋊2Q8 D20.2Q8 C20.65(C4⋊C4) C8×C5⋊D4 C40⋊32D4 Dic5⋊5M4(2) C20.51(C4⋊C4) C40⋊D4 C40⋊18D4 Dic5⋊D8 (C2×D8).D5 Dic5⋊3SD16 Dic5⋊5SD16 (C5×D4).D4 (C5×Q8).D4 Dic5⋊3Q16 (C2×Q16)⋊D5 C60.13Q8 C60.14Q8 C60.26Q8
C20.8Q8 is a maximal quotient of
C20.53D8 C20.39SD16 C40.88D4 C40.9Q8 (C2×C40)⋊15C4 C60.13Q8 C60.14Q8 C60.26Q8
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | D4 | Q8 | D5 | M4(2) | D10 | Dic10 | C5⋊D4 | C4×D5 | C8×D5 | C8⋊D5 |
| kernel | C20.8Q8 | C2×C5⋊2C8 | C4×Dic5 | C2×C40 | C2×Dic5 | Dic5 | C20 | C20 | C2×C8 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C20.8Q8 ►in GL4(𝔽41) generated by
| 13 | 32 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 9 | 19 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 8 | 27 |
| 0 | 0 | 14 | 33 |
| 3 | 18 | 0 | 0 |
| 36 | 38 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 24 | 38 |
G:=sub<GL(4,GF(41))| [13,9,0,0,32,0,0,0,0,0,0,9,0,0,32,19],[3,0,0,0,0,3,0,0,0,0,8,14,0,0,27,33],[3,36,0,0,18,38,0,0,0,0,3,24,0,0,3,38] >;
C20.8Q8 in GAP, Magma, Sage, TeX
C_{20}._8Q_8 % in TeX
G:=Group("C20.8Q8"); // GroupNames label
G:=SmallGroup(160,21);
// by ID
G=gap.SmallGroup(160,21);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,121,31,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^4=a^10,c^2=a^15*b^2,a*b=b*a,c*a*c^-1=a^9,c*b*c^-1=a^15*b^3>;
// generators/relations
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