C80.9C4 - 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)

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

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

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

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

104 conjugacy classes

class	1	2A	2B	4A	4B	4C	5A	5B	8A	8B	8C	8D	8E	8F	10A	···	10F	16A	···	16H	16I	16J	16K	16L	20A	···	20H	32A	···	32P	40A	···	40P	80A	···	80AF
order	1	2	2	4	4	4	5	5	8	8	8	8	8	8	10	···	10	16	···	16	16	16	16	16	20	···	20	32	···	32	40	···	40	80	···	80
size	1	1	2	1	1	2	2	2	1	1	1	1	2	2	2	···	2	1	···	1	2	2	2	2	2	···	2	10	···	10	2	···	2	2	···	2

104 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+							+	-	+	-
image	C₁	C₂	C₂	C₄	C₄	C₈	C₈	C₁₆	C₁₆	D₅	Dic₅	D₁₀	Dic₅	C₅⋊₂C₈	C₅⋊₂C₈	M₆(2)	C₅⋊₂C₁₆	C₅⋊₂C₁₆	C₈₀.₉C₄
kernel	C₈₀.₉C₄	C₅⋊₂C₃₂	C₂×C₈₀	C₈₀	C₂×C₄₀	C₄₀	C₂×C₂₀	C₂₀	C₂×C₁₀	C₂×C₁₆	C₁₆	C₁₆	C₂×C₈	C₈	C₂×C₄	C₅	C₄	C₂²	C₁
# reps	1	2	1	2	2	4	4	8	8	2	2	2	2	4	4	8	8	8	32

Matrix representation of C₈₀.₉C₄ ►in GL₂(𝔽₆₄₁) generated by

433	278
0	196

2	179
441	639

G:=sub<GL(2,GF(641))| [433,0,278,196],[2,441,179,639] >;

C₈₀.₉C₄ in GAP, Magma, Sage, TeX

C_{80}._9C_4

% in TeX

G:=Group("C80.9C4");

// GroupNames label

G:=SmallGroup(320,57);

// by ID

G=gap.SmallGroup(320,57);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,58,80,102,12550]);

// Polycyclic

G:=Group<a,b|a^80=1,b^4=a^50,b*a*b^-1=a^9>;

// generators/relations

Export

Subgroup lattice of C₈₀.₉C₄ in TeX

G = C80.9C4 order 320 = 26·5

4th non-split extension by C80 of C4 acting via C4/C2=C2

G = C₈₀.₉C₄ order 320 = 2⁶·5

4^th non-split extension by C₈₀ of C₄ acting via C₄/C₂=C₂