metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.5D40, C20.30D8, C20.26SD16, C42.261D10, (C4×C8)⋊5D5, (C4×C40)⋊5C2, C2.5(C2×D40), C10.3(C2×D8), C20⋊2Q8⋊2C2, (C2×C4).80D20, C5⋊1(C4.4D8), D20⋊5C4⋊1C2, C20⋊4D4.3C2, (C2×C20).377D4, (C2×C8).287D10, C4.5(C40⋊C2), C10.5(C2×SD16), (C2×D20).4C22, C22.90(C2×D20), C4⋊Dic5.5C22, C20.218(C4○D4), C4.102(C4○D20), (C2×C20).723C23, (C2×C40).347C22, (C4×C20).307C22, C10.5(C4.4D4), C2.10(C4.D20), C2.8(C2×C40⋊C2), (C2×C10).106(C2×D4), (C2×C4).666(C22×D5), SmallGroup(320,321)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.5D40
G = < a,b,c | a4=b40=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 686 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C4⋊C4 [×3], C2×C8 [×2], C2×D4 [×4], C2×Q8, Dic5 [×2], C20 [×6], D10 [×6], C2×C10, C4×C8, D4⋊C4 [×4], C4⋊1D4, C4⋊Q8, C40 [×2], Dic10 [×2], D20 [×8], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4.4D8, C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C2×C40 [×2], C2×Dic10, C2×D20 [×2], C2×D20 [×2], D20⋊5C4 [×4], C4×C40, C20⋊2Q8, C20⋊4D4, C4.5D40
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C2×D8, C2×SD16, D20 [×2], C22×D5, C4.4D8, C40⋊C2 [×2], D40 [×2], C2×D20, C4○D20 [×2], C4.D20, C2×C40⋊C2, C2×D40, C4.5D40
(1 82 41 146)(2 83 42 147)(3 84 43 148)(4 85 44 149)(5 86 45 150)(6 87 46 151)(7 88 47 152)(8 89 48 153)(9 90 49 154)(10 91 50 155)(11 92 51 156)(12 93 52 157)(13 94 53 158)(14 95 54 159)(15 96 55 160)(16 97 56 121)(17 98 57 122)(18 99 58 123)(19 100 59 124)(20 101 60 125)(21 102 61 126)(22 103 62 127)(23 104 63 128)(24 105 64 129)(25 106 65 130)(26 107 66 131)(27 108 67 132)(28 109 68 133)(29 110 69 134)(30 111 70 135)(31 112 71 136)(32 113 72 137)(33 114 73 138)(34 115 74 139)(35 116 75 140)(36 117 76 141)(37 118 77 142)(38 119 78 143)(39 120 79 144)(40 81 80 145)
(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 80 41 40)(2 39 42 79)(3 78 43 38)(4 37 44 77)(5 76 45 36)(6 35 46 75)(7 74 47 34)(8 33 48 73)(9 72 49 32)(10 31 50 71)(11 70 51 30)(12 29 52 69)(13 68 53 28)(14 27 54 67)(15 66 55 26)(16 25 56 65)(17 64 57 24)(18 23 58 63)(19 62 59 22)(20 21 60 61)(81 146 145 82)(83 144 147 120)(84 119 148 143)(85 142 149 118)(86 117 150 141)(87 140 151 116)(88 115 152 139)(89 138 153 114)(90 113 154 137)(91 136 155 112)(92 111 156 135)(93 134 157 110)(94 109 158 133)(95 132 159 108)(96 107 160 131)(97 130 121 106)(98 105 122 129)(99 128 123 104)(100 103 124 127)(101 126 125 102)
G:=sub<Sym(160)| (1,82,41,146)(2,83,42,147)(3,84,43,148)(4,85,44,149)(5,86,45,150)(6,87,46,151)(7,88,47,152)(8,89,48,153)(9,90,49,154)(10,91,50,155)(11,92,51,156)(12,93,52,157)(13,94,53,158)(14,95,54,159)(15,96,55,160)(16,97,56,121)(17,98,57,122)(18,99,58,123)(19,100,59,124)(20,101,60,125)(21,102,61,126)(22,103,62,127)(23,104,63,128)(24,105,64,129)(25,106,65,130)(26,107,66,131)(27,108,67,132)(28,109,68,133)(29,110,69,134)(30,111,70,135)(31,112,71,136)(32,113,72,137)(33,114,73,138)(34,115,74,139)(35,116,75,140)(36,117,76,141)(37,118,77,142)(38,119,78,143)(39,120,79,144)(40,81,80,145), (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), (1,80,41,40)(2,39,42,79)(3,78,43,38)(4,37,44,77)(5,76,45,36)(6,35,46,75)(7,74,47,34)(8,33,48,73)(9,72,49,32)(10,31,50,71)(11,70,51,30)(12,29,52,69)(13,68,53,28)(14,27,54,67)(15,66,55,26)(16,25,56,65)(17,64,57,24)(18,23,58,63)(19,62,59,22)(20,21,60,61)(81,146,145,82)(83,144,147,120)(84,119,148,143)(85,142,149,118)(86,117,150,141)(87,140,151,116)(88,115,152,139)(89,138,153,114)(90,113,154,137)(91,136,155,112)(92,111,156,135)(93,134,157,110)(94,109,158,133)(95,132,159,108)(96,107,160,131)(97,130,121,106)(98,105,122,129)(99,128,123,104)(100,103,124,127)(101,126,125,102)>;
G:=Group( (1,82,41,146)(2,83,42,147)(3,84,43,148)(4,85,44,149)(5,86,45,150)(6,87,46,151)(7,88,47,152)(8,89,48,153)(9,90,49,154)(10,91,50,155)(11,92,51,156)(12,93,52,157)(13,94,53,158)(14,95,54,159)(15,96,55,160)(16,97,56,121)(17,98,57,122)(18,99,58,123)(19,100,59,124)(20,101,60,125)(21,102,61,126)(22,103,62,127)(23,104,63,128)(24,105,64,129)(25,106,65,130)(26,107,66,131)(27,108,67,132)(28,109,68,133)(29,110,69,134)(30,111,70,135)(31,112,71,136)(32,113,72,137)(33,114,73,138)(34,115,74,139)(35,116,75,140)(36,117,76,141)(37,118,77,142)(38,119,78,143)(39,120,79,144)(40,81,80,145), (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), (1,80,41,40)(2,39,42,79)(3,78,43,38)(4,37,44,77)(5,76,45,36)(6,35,46,75)(7,74,47,34)(8,33,48,73)(9,72,49,32)(10,31,50,71)(11,70,51,30)(12,29,52,69)(13,68,53,28)(14,27,54,67)(15,66,55,26)(16,25,56,65)(17,64,57,24)(18,23,58,63)(19,62,59,22)(20,21,60,61)(81,146,145,82)(83,144,147,120)(84,119,148,143)(85,142,149,118)(86,117,150,141)(87,140,151,116)(88,115,152,139)(89,138,153,114)(90,113,154,137)(91,136,155,112)(92,111,156,135)(93,134,157,110)(94,109,158,133)(95,132,159,108)(96,107,160,131)(97,130,121,106)(98,105,122,129)(99,128,123,104)(100,103,124,127)(101,126,125,102) );
G=PermutationGroup([(1,82,41,146),(2,83,42,147),(3,84,43,148),(4,85,44,149),(5,86,45,150),(6,87,46,151),(7,88,47,152),(8,89,48,153),(9,90,49,154),(10,91,50,155),(11,92,51,156),(12,93,52,157),(13,94,53,158),(14,95,54,159),(15,96,55,160),(16,97,56,121),(17,98,57,122),(18,99,58,123),(19,100,59,124),(20,101,60,125),(21,102,61,126),(22,103,62,127),(23,104,63,128),(24,105,64,129),(25,106,65,130),(26,107,66,131),(27,108,67,132),(28,109,68,133),(29,110,69,134),(30,111,70,135),(31,112,71,136),(32,113,72,137),(33,114,73,138),(34,115,74,139),(35,116,75,140),(36,117,76,141),(37,118,77,142),(38,119,78,143),(39,120,79,144),(40,81,80,145)], [(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)], [(1,80,41,40),(2,39,42,79),(3,78,43,38),(4,37,44,77),(5,76,45,36),(6,35,46,75),(7,74,47,34),(8,33,48,73),(9,72,49,32),(10,31,50,71),(11,70,51,30),(12,29,52,69),(13,68,53,28),(14,27,54,67),(15,66,55,26),(16,25,56,65),(17,64,57,24),(18,23,58,63),(19,62,59,22),(20,21,60,61),(81,146,145,82),(83,144,147,120),(84,119,148,143),(85,142,149,118),(86,117,150,141),(87,140,151,116),(88,115,152,139),(89,138,153,114),(90,113,154,137),(91,136,155,112),(92,111,156,135),(93,134,157,110),(94,109,158,133),(95,132,159,108),(96,107,160,131),(97,130,121,106),(98,105,122,129),(99,128,123,104),(100,103,124,127),(101,126,125,102)])
86 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
size | 1 | 1 | 1 | 1 | 40 | 40 | 2 | ··· | 2 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
image | C1 | C2 | C2 | C2 | C2 | D4 | D5 | D8 | SD16 | C4○D4 | D10 | D10 | D20 | C40⋊C2 | D40 | C4○D20 |
kernel | C4.5D40 | D20⋊5C4 | C4×C40 | C20⋊2Q8 | C20⋊4D4 | C2×C20 | C4×C8 | C20 | C20 | C20 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 |
# reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C4.5D40 ►in GL4(𝔽41) generated by
30 | 28 | 0 | 0 |
22 | 11 | 0 | 0 |
0 | 0 | 40 | 0 |
0 | 0 | 0 | 40 |
22 | 13 | 0 | 0 |
19 | 0 | 0 | 0 |
0 | 0 | 0 | 11 |
0 | 0 | 26 | 17 |
19 | 32 | 0 | 0 |
22 | 22 | 0 | 0 |
0 | 0 | 0 | 11 |
0 | 0 | 15 | 0 |
G:=sub<GL(4,GF(41))| [30,22,0,0,28,11,0,0,0,0,40,0,0,0,0,40],[22,19,0,0,13,0,0,0,0,0,0,26,0,0,11,17],[19,22,0,0,32,22,0,0,0,0,0,15,0,0,11,0] >;
C4.5D40 in GAP, Magma, Sage, TeX
C_4._5D_{40}
% in TeX
G:=Group("C4.5D40");
// GroupNames label
G:=SmallGroup(320,321);
// by ID
G=gap.SmallGroup(320,321);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^4=b^40=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations