C80.2C4 - 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)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4
type	+	+	+			-	+	+	-		+	+
image	C₁	C₂	C₂	C₄	C₄	Q₈	D₄	D₈	Q₁₆	C₈.₄Q₈	F₅	C₂×F₅	C₄⋊F₅	D₅.D₈	C₈₀.₂C₄
kernel	C₈₀.₂C₄	D₅×C₁₆	D₁₀.Q₈	C₅⋊₂C₁₆	C₈₀	C₅⋊₂C₈	C₄×D₅	Dic₅	D₁₀	C₅	C₁₆	C₈	C₄	C₂	C₁
# reps	1	1	2	2	2	1	1	2	2	8	1	1	2	4	8

Matrix representation of C₈₀.₂C₄ ►in GL₄(𝔽₇) generated by

2	6	1	6
5	6	1	4
2	6	0	2
2	0	2	4

1	3	5	4
0	4	4	3
2	3	3	3
4	1	4	6

G:=sub<GL(4,GF(7))| [2,5,2,2,6,6,6,0,1,1,0,2,6,4,2,4],[1,0,2,4,3,4,3,1,5,4,3,4,4,3,3,6] >;

C₈₀.₂C₄ in GAP, Magma, Sage, TeX

C_{80}._2C_4

% in TeX

G:=Group("C80.2C4");

// GroupNames label

G:=SmallGroup(320,190);

// by ID

G=gap.SmallGroup(320,190);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,184,1571,80,1684,102,6278,3156]);

// Polycyclic

G:=Group<a,b|a^80=1,b^4=a^40,b*a*b^-1=a^23>;

// generators/relations

Export

Subgroup lattice of C₈₀.₂C₄ in TeX

G = C80.2C4 order 320 = 26·5

2nd non-split extension by C80 of C4 acting faithfully

G = C₈₀.₂C₄ order 320 = 2⁶·5

2^nd non-split extension by C₈₀ of C₄ acting faithfully