Dic40 - GroupNames

(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)

(1 111 41 151)(2 110 42 150)(3 109 43 149)(4 108 44 148)(5 107 45 147)(6 106 46 146)(7 105 47 145)(8 104 48 144)(9 103 49 143)(10 102 50 142)(11 101 51 141)(12 100 52 140)(13 99 53 139)(14 98 54 138)(15 97 55 137)(16 96 56 136)(17 95 57 135)(18 94 58 134)(19 93 59 133)(20 92 60 132)(21 91 61 131)(22 90 62 130)(23 89 63 129)(24 88 64 128)(25 87 65 127)(26 86 66 126)(27 85 67 125)(28 84 68 124)(29 83 69 123)(30 82 70 122)(31 81 71 121)(32 160 72 120)(33 159 73 119)(34 158 74 118)(35 157 75 117)(36 156 76 116)(37 155 77 115)(38 154 78 114)(39 153 79 113)(40 152 80 112)

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), (1,111,41,151)(2,110,42,150)(3,109,43,149)(4,108,44,148)(5,107,45,147)(6,106,46,146)(7,105,47,145)(8,104,48,144)(9,103,49,143)(10,102,50,142)(11,101,51,141)(12,100,52,140)(13,99,53,139)(14,98,54,138)(15,97,55,137)(16,96,56,136)(17,95,57,135)(18,94,58,134)(19,93,59,133)(20,92,60,132)(21,91,61,131)(22,90,62,130)(23,89,63,129)(24,88,64,128)(25,87,65,127)(26,86,66,126)(27,85,67,125)(28,84,68,124)(29,83,69,123)(30,82,70,122)(31,81,71,121)(32,160,72,120)(33,159,73,119)(34,158,74,118)(35,157,75,117)(36,156,76,116)(37,155,77,115)(38,154,78,114)(39,153,79,113)(40,152,80,112)>;

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), (1,111,41,151)(2,110,42,150)(3,109,43,149)(4,108,44,148)(5,107,45,147)(6,106,46,146)(7,105,47,145)(8,104,48,144)(9,103,49,143)(10,102,50,142)(11,101,51,141)(12,100,52,140)(13,99,53,139)(14,98,54,138)(15,97,55,137)(16,96,56,136)(17,95,57,135)(18,94,58,134)(19,93,59,133)(20,92,60,132)(21,91,61,131)(22,90,62,130)(23,89,63,129)(24,88,64,128)(25,87,65,127)(26,86,66,126)(27,85,67,125)(28,84,68,124)(29,83,69,123)(30,82,70,122)(31,81,71,121)(32,160,72,120)(33,159,73,119)(34,158,74,118)(35,157,75,117)(36,156,76,116)(37,155,77,115)(38,154,78,114)(39,153,79,113)(40,152,80,112) );

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)], [(1,111,41,151),(2,110,42,150),(3,109,43,149),(4,108,44,148),(5,107,45,147),(6,106,46,146),(7,105,47,145),(8,104,48,144),(9,103,49,143),(10,102,50,142),(11,101,51,141),(12,100,52,140),(13,99,53,139),(14,98,54,138),(15,97,55,137),(16,96,56,136),(17,95,57,135),(18,94,58,134),(19,93,59,133),(20,92,60,132),(21,91,61,131),(22,90,62,130),(23,89,63,129),(24,88,64,128),(25,87,65,127),(26,86,66,126),(27,85,67,125),(28,84,68,124),(29,83,69,123),(30,82,70,122),(31,81,71,121),(32,160,72,120),(33,159,73,119),(34,158,74,118),(35,157,75,117),(36,156,76,116),(37,155,77,115),(38,154,78,114),(39,153,79,113),(40,152,80,112)]])

Dic₄₀ is a maximal subgroup of
C₁₆₀⋊C₂ Dic₈₀ D₁₆.D₅ C₅⋊Q₆₄ D₈₀⋊₇C₂ C₁₆.D₁₀ D₁₆⋊₃D₅ SD₃₂⋊D₅ D₅×Q₃₂ C₃⋊Dic₄₀ Dic₁₂₀
Dic₄₀ is a maximal quotient of
C₄₀.₇₈D₄ C₈₀⋊₁₃C₄ C₃⋊Dic₄₀ Dic₁₂₀

43 conjugacy classes

class	1	2	4A	4B	4C	5A	5B	8A	8B	10A	10B	16A	16B	16C	16D	20A	20B	20C	20D	40A	···	40H	80A	···	80P
order	1	2	4	4	4	5	5	8	8	10	10	16	16	16	16	20	20	20	20	40	···	40	80	···	80
size	1	1	2	40	40	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	···	2	2	···	2

43 irreducible representations

dim	1	1	1	2	2	2	2	2	2	2	2
type	+	+	+	+	+	+	+	-	+	+	-
image	C₁	C₂	C₂	D₄	D₅	D₈	D₁₀	Q₃₂	D₂₀	D₄₀	Dic₄₀
kernel	Dic₄₀	C₈₀	Dic₂₀	C₂₀	C₁₆	C₁₀	C₈	C₅	C₄	C₂	C₁
# reps	1	1	2	1	2	2	2	4	4	8	16

Matrix representation of Dic₄₀ ►in GL₂(𝔽₂₄₁) generated by

239	58
183	115

148	42
58	93

G:=sub<GL(2,GF(241))| [239,183,58,115],[148,58,42,93] >;

Dic₄₀ in GAP, Magma, Sage, TeX

{\rm Dic}_{40}

% in TeX

G:=Group("Dic40");

// GroupNames label

G:=SmallGroup(160,8);

// by ID

G=gap.SmallGroup(160,8);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,73,79,218,122,579,69,4613]);

// Polycyclic

G:=Group<a,b|a^80=1,b^2=a^40,b*a*b^-1=a^-1>;

// generators/relations

Export

Subgroup lattice of Dic₄₀ in TeX

G = Dic40 order 160 = 25·5

Dicyclic group

G = Dic₄₀ order 160 = 2⁵·5