metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊1M4(2), C5⋊C8⋊2D4, C4⋊C4.6F5, C5⋊2(C8⋊6D4), C20⋊C8⋊4C2, C4⋊1(C4.F5), C2.12(D4×F5), C10.10(C4×D4), (C2×D20).12C4, D10⋊C8⋊10C2, C2.5(Q8.F5), D10⋊C4.3C4, C10.21(C8○D4), Dic5.71(C2×D4), D20⋊8C4.18C2, C10.14(C2×M4(2)), Dic5.56(C4○D4), C22.77(C22×F5), (C4×Dic5).192C22, (C2×Dic5).332C23, (C4×C5⋊C8)⋊4C2, (C5×C4⋊C4).9C4, C2.9(C2×C4.F5), (C2×C4.F5)⋊12C2, (C2×C4).26(C2×F5), (C2×C20).83(C2×C4), (C2×C5⋊C8).29C22, (C2×C4×D5).277C22, (C2×C10).43(C22×C4), (C2×Dic5).58(C2×C4), (C22×D5).49(C2×C4), SmallGroup(320,1043)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 474 in 122 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×5], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×6], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C5⋊C8 [×2], C5⋊C8 [×3], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C8⋊6D4, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C4.F5 [×4], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C4×D5 [×2], C2×D20, C4×C5⋊C8, C20⋊C8, D10⋊C8 [×2], D20⋊8C4, C2×C4.F5 [×2], C20⋊M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C8⋊6D4, C4.F5 [×2], C22×F5, C2×C4.F5, D4×F5, Q8.F5, C20⋊M4(2)
Generators and relations
G = < a,b,c | a20=b8=c2=1, bab-1=a7, cac=a-1, cbc=b5 >
►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 47 142 85 30 127 118 70)(2 50 151 92 31 130 107 77)(3 53 160 99 32 133 116 64)(4 56 149 86 33 136 105 71)(5 59 158 93 34 139 114 78)(6 42 147 100 35 122 103 65)(7 45 156 87 36 125 112 72)(8 48 145 94 37 128 101 79)(9 51 154 81 38 131 110 66)(10 54 143 88 39 134 119 73)(11 57 152 95 40 137 108 80)(12 60 141 82 21 140 117 67)(13 43 150 89 22 123 106 74)(14 46 159 96 23 126 115 61)(15 49 148 83 24 129 104 68)(16 52 157 90 25 132 113 75)(17 55 146 97 26 135 102 62)(18 58 155 84 27 138 111 69)(19 41 144 91 28 121 120 76)(20 44 153 98 29 124 109 63)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 80)(76 79)(77 78)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 144)(102 143)(103 142)(104 141)(105 160)(106 159)(107 158)(108 157)(109 156)(110 155)(111 154)(112 153)(113 152)(114 151)(115 150)(116 149)(117 148)(118 147)(119 146)(120 145)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)
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), (1,47,142,85,30,127,118,70)(2,50,151,92,31,130,107,77)(3,53,160,99,32,133,116,64)(4,56,149,86,33,136,105,71)(5,59,158,93,34,139,114,78)(6,42,147,100,35,122,103,65)(7,45,156,87,36,125,112,72)(8,48,145,94,37,128,101,79)(9,51,154,81,38,131,110,66)(10,54,143,88,39,134,119,73)(11,57,152,95,40,137,108,80)(12,60,141,82,21,140,117,67)(13,43,150,89,22,123,106,74)(14,46,159,96,23,126,115,61)(15,49,148,83,24,129,104,68)(16,52,157,90,25,132,113,75)(17,55,146,97,26,135,102,62)(18,58,155,84,27,138,111,69)(19,41,144,91,28,121,120,76)(20,44,153,98,29,124,109,63), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,80)(76,79)(77,78)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,144)(102,143)(103,142)(104,141)(105,160)(106,159)(107,158)(108,157)(109,156)(110,155)(111,154)(112,153)(113,152)(114,151)(115,150)(116,149)(117,148)(118,147)(119,146)(120,145)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)>;
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), (1,47,142,85,30,127,118,70)(2,50,151,92,31,130,107,77)(3,53,160,99,32,133,116,64)(4,56,149,86,33,136,105,71)(5,59,158,93,34,139,114,78)(6,42,147,100,35,122,103,65)(7,45,156,87,36,125,112,72)(8,48,145,94,37,128,101,79)(9,51,154,81,38,131,110,66)(10,54,143,88,39,134,119,73)(11,57,152,95,40,137,108,80)(12,60,141,82,21,140,117,67)(13,43,150,89,22,123,106,74)(14,46,159,96,23,126,115,61)(15,49,148,83,24,129,104,68)(16,52,157,90,25,132,113,75)(17,55,146,97,26,135,102,62)(18,58,155,84,27,138,111,69)(19,41,144,91,28,121,120,76)(20,44,153,98,29,124,109,63), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,80)(76,79)(77,78)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,144)(102,143)(103,142)(104,141)(105,160)(106,159)(107,158)(108,157)(109,156)(110,155)(111,154)(112,153)(113,152)(114,151)(115,150)(116,149)(117,148)(118,147)(119,146)(120,145)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135) );
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)], [(1,47,142,85,30,127,118,70),(2,50,151,92,31,130,107,77),(3,53,160,99,32,133,116,64),(4,56,149,86,33,136,105,71),(5,59,158,93,34,139,114,78),(6,42,147,100,35,122,103,65),(7,45,156,87,36,125,112,72),(8,48,145,94,37,128,101,79),(9,51,154,81,38,131,110,66),(10,54,143,88,39,134,119,73),(11,57,152,95,40,137,108,80),(12,60,141,82,21,140,117,67),(13,43,150,89,22,123,106,74),(14,46,159,96,23,126,115,61),(15,49,148,83,24,129,104,68),(16,52,157,90,25,132,113,75),(17,55,146,97,26,135,102,62),(18,58,155,84,27,138,111,69),(19,41,144,91,28,121,120,76),(20,44,153,98,29,124,109,63)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,40),(17,39),(18,38),(19,37),(20,36),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(75,80),(76,79),(77,78),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,144),(102,143),(103,142),(104,141),(105,160),(106,159),(107,158),(108,157),(109,156),(110,155),(111,154),(112,153),(113,152),(114,151),(115,150),(116,149),(117,148),(118,147),(119,146),(120,145),(121,128),(122,127),(123,126),(124,125),(129,140),(130,139),(131,138),(132,137),(133,136),(134,135)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 8 | 2 | 2 |
| 0 | 0 | 23 | 16 | 23 | 10 |
| 0 | 0 | 31 | 18 | 25 | 18 |
| 0 | 0 | 39 | 39 | 33 | 20 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,21,23,31,39,0,0,8,16,18,39,0,0,2,23,25,33,0,0,2,10,18,20],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | C4.F5 | D4×F5 | Q8.F5 |
| kernel | C20⋊M4(2) | C4×C5⋊C8 | C20⋊C8 | D10⋊C8 | D20⋊8C4 | C2×C4.F5 | D10⋊C4 | C5×C4⋊C4 | C2×D20 | C5⋊C8 | Dic5 | C20 | C10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_{20}\rtimes M_{4(2)} % in TeX
G:=Group("C20:M4(2)"); // GroupNames label
G:=SmallGroup(320,1043);
// by ID
G=gap.SmallGroup(320,1043);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations
×
⋊
ℤ
𝔽
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