metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊1C8, (C22×C4).5F5, C23.31(C2×F5), C10.9(C23⋊C4), (C22×C20).10C4, C10.11(C22⋊C8), (C2×Dic5).101D4, (C2×C10).19M4(2), (C22×Dic5).6C4, C23.2F5.1C2, C2.1(D10.D4), C10.3(C4.10D4), C22.35(C22⋊F5), C2.4(C23.2F5), C22.4(C22.F5), C2.1(Dic5.D4), C5⋊1(C22.M4(2)), (C22×Dic5).171C22, (C2×C4)⋊(C5⋊C8), C22.3(C2×C5⋊C8), (C2×C10).28(C2×C8), (C2×C4⋊Dic5).2C2, (C22×C10).42(C2×C4), (C2×C10).26(C22⋊C4), SmallGroup(320,251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20)⋊1C8
G = < a,b,c | a2=b20=c8=1, ab=ba, cac-1=ab10, cbc-1=ab3 >
Subgroups: 306 in 78 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C22⋊C8, C2×C4⋊C4, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22.M4(2), C4⋊Dic5, C2×C5⋊C8, C22×Dic5, C22×C20, C23.2F5, C2×C4⋊Dic5, (C2×C20)⋊1C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.10D4, C5⋊C8, C2×F5, C22.M4(2), C2×C5⋊C8, C22.F5, C22⋊F5, D10.D4, Dic5.D4, C23.2F5, (C2×C20)⋊1C8
►On 160 points
Generators in S160
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(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)
(1 129 94 60 152 114 70 31)(2 136 93 53 153 101 69 24)(3 123 92 46 154 108 68 37)(4 130 91 59 155 115 67 30)(5 137 90 52 156 102 66 23)(6 124 89 45 157 109 65 36)(7 131 88 58 158 116 64 29)(8 138 87 51 159 103 63 22)(9 125 86 44 160 110 62 35)(10 132 85 57 141 117 61 28)(11 139 84 50 142 104 80 21)(12 126 83 43 143 111 79 34)(13 133 82 56 144 118 78 27)(14 140 81 49 145 105 77 40)(15 127 100 42 146 112 76 33)(16 134 99 55 147 119 75 26)(17 121 98 48 148 106 74 39)(18 128 97 41 149 113 73 32)(19 135 96 54 150 120 72 25)(20 122 95 47 151 107 71 38)
G:=sub<Sym(160)| (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (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), (1,129,94,60,152,114,70,31)(2,136,93,53,153,101,69,24)(3,123,92,46,154,108,68,37)(4,130,91,59,155,115,67,30)(5,137,90,52,156,102,66,23)(6,124,89,45,157,109,65,36)(7,131,88,58,158,116,64,29)(8,138,87,51,159,103,63,22)(9,125,86,44,160,110,62,35)(10,132,85,57,141,117,61,28)(11,139,84,50,142,104,80,21)(12,126,83,43,143,111,79,34)(13,133,82,56,144,118,78,27)(14,140,81,49,145,105,77,40)(15,127,100,42,146,112,76,33)(16,134,99,55,147,119,75,26)(17,121,98,48,148,106,74,39)(18,128,97,41,149,113,73,32)(19,135,96,54,150,120,72,25)(20,122,95,47,151,107,71,38)>;
G:=Group( (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140), (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), (1,129,94,60,152,114,70,31)(2,136,93,53,153,101,69,24)(3,123,92,46,154,108,68,37)(4,130,91,59,155,115,67,30)(5,137,90,52,156,102,66,23)(6,124,89,45,157,109,65,36)(7,131,88,58,158,116,64,29)(8,138,87,51,159,103,63,22)(9,125,86,44,160,110,62,35)(10,132,85,57,141,117,61,28)(11,139,84,50,142,104,80,21)(12,126,83,43,143,111,79,34)(13,133,82,56,144,118,78,27)(14,140,81,49,145,105,77,40)(15,127,100,42,146,112,76,33)(16,134,99,55,147,119,75,26)(17,121,98,48,148,106,74,39)(18,128,97,41,149,113,73,32)(19,135,96,54,150,120,72,25)(20,122,95,47,151,107,71,38) );
G=PermutationGroup([[(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140)], [(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)], [(1,129,94,60,152,114,70,31),(2,136,93,53,153,101,69,24),(3,123,92,46,154,108,68,37),(4,130,91,59,155,115,67,30),(5,137,90,52,156,102,66,23),(6,124,89,45,157,109,65,36),(7,131,88,58,158,116,64,29),(8,138,87,51,159,103,63,22),(9,125,86,44,160,110,62,35),(10,132,85,57,141,117,61,28),(11,139,84,50,142,104,80,21),(12,126,83,43,143,111,79,34),(13,133,82,56,144,118,78,27),(14,140,81,49,145,105,77,40),(15,127,100,42,146,112,76,33),(16,134,99,55,147,119,75,26),(17,121,98,48,148,106,74,39),(18,128,97,41,149,113,73,32),(19,135,96,54,150,120,72,25),(20,122,95,47,151,107,71,38)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 4 | 20 | ··· | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | F5 | C23⋊C4 | C4.10D4 | C5⋊C8 | C2×F5 | C22.F5 | C22⋊F5 | D10.D4 | Dic5.D4 |
| kernel | (C2×C20)⋊1C8 | C23.2F5 | C2×C4⋊Dic5 | C22×Dic5 | C22×C20 | C2×C20 | C2×Dic5 | C2×C10 | C22×C4 | C10 | C10 | C2×C4 | C23 | C22 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of (C2×C20)⋊1C8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 39 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 16 |
| 0 | 0 | 0 | 0 | 25 | 39 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 25 | 39 | 0 | 0 |
| 0 | 0 | 25 | 16 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,25,2,0,0,0,0,39,13,0,0,0,0,0,0,25,25,0,0,0,0,16,39],[0,38,0,0,0,0,3,0,0,0,0,0,0,0,0,0,25,25,0,0,0,0,39,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
(C2×C20)⋊1C8 in GAP, Magma, Sage, TeX
(C_2\times C_{20})\rtimes_1C_8 % in TeX
G:=Group("(C2xC20):1C8"); // GroupNames label
G:=SmallGroup(320,251);
// by ID
G=gap.SmallGroup(320,251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,100,1123,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^2=b^20=c^8=1,a*b=b*a,c*a*c^-1=a*b^10,c*b*c^-1=a*b^3>;
// generators/relations