C5:2C32 - GroupNames

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

(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)

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

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

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

C₅⋊₂C₃₂ is a maximal subgroup of
C₅⋊C₆₄ D₅×C₃₂ C₃₂⋊D₅ C₈₀.₉C₄ C₅⋊D₃₂ D₁₆.D₅ C₅⋊SD₆₄ C₅⋊Q₆₄ C₁₅⋊₃C₃₂
C₅⋊₂C₃₂ is a maximal quotient of
C₅⋊₂C₆₄ C₁₅⋊₃C₃₂

64 conjugacy classes

class	1	2	4A	4B	5A	5B	8A	8B	8C	8D	10A	10B	16A	···	16H	20A	20B	20C	20D	32A	···	32P	40A	···	40H	80A	···	80P
order	1	2	4	4	5	5	8	8	8	8	10	10	16	···	16	20	20	20	20	32	···	32	40	···	40	80	···	80
size	1	1	1	1	2	2	1	1	1	1	2	2	1	···	1	2	2	2	2	5	···	5	2	···	2	2	···	2

64 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2
type	+	+					+	-
image	C₁	C₂	C₄	C₈	C₁₆	C₃₂	D₅	Dic₅	C₅⋊₂C₈	C₅⋊₂C₁₆	C₅⋊₂C₃₂
kernel	C₅⋊₂C₃₂	C₈₀	C₄₀	C₂₀	C₁₀	C₅	C₁₆	C₈	C₄	C₂	C₁
# reps	1	1	2	4	8	16	2	2	4	8	16

Matrix representation of C₅⋊₂C₃₂ ►in GL₂(𝔽₆₄₁) generated by

640	1
361	279

500	94
301	141

G:=sub<GL(2,GF(641))| [640,361,1,279],[500,301,94,141] >;

C₅⋊₂C₃₂ in GAP, Magma, Sage, TeX

C_5\rtimes_2C_{32}

% in TeX

G:=Group("C5:2C32");

// GroupNames label

G:=SmallGroup(160,1);

// by ID

G=gap.SmallGroup(160,1);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,12,31,50,69,4613]);

// Polycyclic

G:=Group<a,b|a^5=b^32=1,b*a*b^-1=a^-1>;

// generators/relations

Export

Subgroup lattice of C₅⋊₂C₃₂ in TeX

G = C5⋊2C32 order 160 = 25·5

The semidirect product of C5 and C32 acting via C32/C16=C2

G = C₅⋊₂C₃₂ order 160 = 2⁵·5

The semidirect product of C₅ and C₃₂ acting via C₃₂/C₁₆=C₂