Dic42 - 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)

(1 86 43 128)(2 85 44 127)(3 168 45 126)(4 167 46 125)(5 166 47 124)(6 165 48 123)(7 164 49 122)(8 163 50 121)(9 162 51 120)(10 161 52 119)(11 160 53 118)(12 159 54 117)(13 158 55 116)(14 157 56 115)(15 156 57 114)(16 155 58 113)(17 154 59 112)(18 153 60 111)(19 152 61 110)(20 151 62 109)(21 150 63 108)(22 149 64 107)(23 148 65 106)(24 147 66 105)(25 146 67 104)(26 145 68 103)(27 144 69 102)(28 143 70 101)(29 142 71 100)(30 141 72 99)(31 140 73 98)(32 139 74 97)(33 138 75 96)(34 137 76 95)(35 136 77 94)(36 135 78 93)(37 134 79 92)(38 133 80 91)(39 132 81 90)(40 131 82 89)(41 130 83 88)(42 129 84 87)

G:=sub<Sym(168)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,86,43,128)(2,85,44,127)(3,168,45,126)(4,167,46,125)(5,166,47,124)(6,165,48,123)(7,164,49,122)(8,163,50,121)(9,162,51,120)(10,161,52,119)(11,160,53,118)(12,159,54,117)(13,158,55,116)(14,157,56,115)(15,156,57,114)(16,155,58,113)(17,154,59,112)(18,153,60,111)(19,152,61,110)(20,151,62,109)(21,150,63,108)(22,149,64,107)(23,148,65,106)(24,147,66,105)(25,146,67,104)(26,145,68,103)(27,144,69,102)(28,143,70,101)(29,142,71,100)(30,141,72,99)(31,140,73,98)(32,139,74,97)(33,138,75,96)(34,137,76,95)(35,136,77,94)(36,135,78,93)(37,134,79,92)(38,133,80,91)(39,132,81,90)(40,131,82,89)(41,130,83,88)(42,129,84,87)>;

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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,86,43,128)(2,85,44,127)(3,168,45,126)(4,167,46,125)(5,166,47,124)(6,165,48,123)(7,164,49,122)(8,163,50,121)(9,162,51,120)(10,161,52,119)(11,160,53,118)(12,159,54,117)(13,158,55,116)(14,157,56,115)(15,156,57,114)(16,155,58,113)(17,154,59,112)(18,153,60,111)(19,152,61,110)(20,151,62,109)(21,150,63,108)(22,149,64,107)(23,148,65,106)(24,147,66,105)(25,146,67,104)(26,145,68,103)(27,144,69,102)(28,143,70,101)(29,142,71,100)(30,141,72,99)(31,140,73,98)(32,139,74,97)(33,138,75,96)(34,137,76,95)(35,136,77,94)(36,135,78,93)(37,134,79,92)(38,133,80,91)(39,132,81,90)(40,131,82,89)(41,130,83,88)(42,129,84,87) );

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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,86,43,128),(2,85,44,127),(3,168,45,126),(4,167,46,125),(5,166,47,124),(6,165,48,123),(7,164,49,122),(8,163,50,121),(9,162,51,120),(10,161,52,119),(11,160,53,118),(12,159,54,117),(13,158,55,116),(14,157,56,115),(15,156,57,114),(16,155,58,113),(17,154,59,112),(18,153,60,111),(19,152,61,110),(20,151,62,109),(21,150,63,108),(22,149,64,107),(23,148,65,106),(24,147,66,105),(25,146,67,104),(26,145,68,103),(27,144,69,102),(28,143,70,101),(29,142,71,100),(30,141,72,99),(31,140,73,98),(32,139,74,97),(33,138,75,96),(34,137,76,95),(35,136,77,94),(36,135,78,93),(37,134,79,92),(38,133,80,91),(39,132,81,90),(40,131,82,89),(41,130,83,88),(42,129,84,87)]])

Dic₄₂ is a maximal subgroup of
C₆.D₂₈ D₁₂.D₇ C₃⋊Dic₂₈ C₇⋊Dic₁₂ C₈⋊D₂₁ Dic₈₄ D₄.D₂₁ C₂₁⋊₇Q₁₆ D₇×Dic₆ D₂₈⋊₅S₃ S₃×Dic₁₄ D₁₂⋊₅D₇ D₈₄⋊₁₁C₂ D₄⋊₂D₂₁ Q₈×D₂₁
Dic₄₂ is a maximal quotient of
C₄₂.₄Q₈ C₈₄⋊C₄

45 conjugacy classes

class	1	2	3	4A	4B	4C	6	7A	7B	7C	12A	12B	14A	14B	14C	21A	···	21F	28A	···	28F	42A	···	42F	84A	···	84L
order	1	2	3	4	4	4	6	7	7	7	12	12	14	14	14	21	···	21	28	···	28	42	···	42	84	···	84
size	1	1	2	2	42	42	2	2	2	2	2	2	2	2	2	2	···	2	2	···	2	2	···	2	2	···	2

45 irreducible representations

dim	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+	-	+	+	-	+	+	-	+	-
image	C₁	C₂	C₂	S₃	Q₈	D₆	D₇	Dic₆	D₁₄	D₂₁	Dic₁₄	D₄₂	Dic₄₂
kernel	Dic₄₂	Dic₂₁	C₈₄	C₂₈	C₂₁	C₁₄	C₁₂	C₇	C₆	C₄	C₃	C₂	C₁
# reps	1	2	1	1	1	1	3	2	3	6	6	6	12

Matrix representation of Dic₄₂ ►in GL₂(𝔽₃₃₇) generated by

144	120
133	277

126	323
99	211

G:=sub<GL(2,GF(337))| [144,133,120,277],[126,99,323,211] >;

Dic₄₂ in GAP, Magma, Sage, TeX

{\rm Dic}_{42}

% in TeX

G:=Group("Dic42");

// GroupNames label

G:=SmallGroup(168,34);

// by ID

G=gap.SmallGroup(168,34);

# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,20,61,26,323,3604]);

// Polycyclic

G:=Group<a,b|a^84=1,b^2=a^42,b*a*b^-1=a^-1>;

// generators/relations

Export

Subgroup lattice of Dic₄₂ in TeX

G = Dic42 order 168 = 23·3·7

Dicyclic group

G = Dic₄₂ order 168 = 2³·3·7