C80.6C4 - 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 96 61 116 41 136 21 156)(2 135 62 155 42 95 22 115)(3 94 63 114 43 134 23 154)(4 133 64 153 44 93 24 113)(5 92 65 112 45 132 25 152)(6 131 66 151 46 91 26 111)(7 90 67 110 47 130 27 150)(8 129 68 149 48 89 28 109)(9 88 69 108 49 128 29 148)(10 127 70 147 50 87 30 107)(11 86 71 106 51 126 31 146)(12 125 72 145 52 85 32 105)(13 84 73 104 53 124 33 144)(14 123 74 143 54 83 34 103)(15 82 75 102 55 122 35 142)(16 121 76 141 56 81 36 101)(17 160 77 100 57 120 37 140)(18 119 78 139 58 159 38 99)(19 158 79 98 59 118 39 138)(20 117 80 137 60 157 40 97)

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

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

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

86 conjugacy classes

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

86 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+		-	+	+	+	-	-	+	-	+		+	-
image	C₁	C₂	C₂	C₄	Q₈	D₄	D₅	D₈	Q₁₆	Dic₅	D₁₀	Dic₁₀	D₂₀	C₈.₄Q₈	D₄₀	Dic₂₀	C₈₀.₆C₄
kernel	C₈₀.₆C₄	C₄₀.₆C₄	C₂×C₈₀	C₈₀	C₄₀	C₂×C₂₀	C₂×C₁₆	C₂₀	C₂×C₁₀	C₁₆	C₂×C₈	C₈	C₂×C₄	C₅	C₄	C₂²	C₁
# reps	1	2	1	4	1	1	2	2	2	4	2	4	4	8	8	8	32

Matrix representation of C₈₀.₆C₄ ►in GL₂(𝔽₂₄₁) generated by

57	0
99	93

77	28
109	164

G:=sub<GL(2,GF(241))| [57,99,0,93],[77,109,28,164] >;

C₈₀.₆C₄ in GAP, Magma, Sage, TeX

C_{80}._6C_4

% in TeX

G:=Group("C80.6C4");

// GroupNames label

G:=SmallGroup(320,64);

// by ID

G=gap.SmallGroup(320,64);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,675,192,1684,102,12550]);

// Polycyclic

G:=Group<a,b|a^80=1,b^4=a^40,b*a*b^-1=a^39>;

// generators/relations

Export

Subgroup lattice of C₈₀.₆C₄ in TeX

G = C80.6C4 order 320 = 26·5

1st non-split extension by C80 of C4 acting via C4/C2=C2

G = C₈₀.₆C₄ order 320 = 2⁶·5

1^st non-split extension by C₈₀ of C₄ acting via C₄/C₂=C₂