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## G = C5×M6(2)  order 320 = 26·5

### Direct product of C5 and M6(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×M6(2)
 Chief series C1 — C2 — C4 — C8 — C16 — C80 — C160 — C5×M6(2)
 Lower central C1 — C2 — C5×M6(2)
 Upper central C1 — C80 — C5×M6(2)

Generators and relations for C5×M6(2)
G = < a,b,c | a5=b32=c2=1, ab=ba, ac=ca, cbc=b17 >

Smallest permutation representation of C5×M6(2)
On 160 points
Generators in S160
(1 136 105 50 83)(2 137 106 51 84)(3 138 107 52 85)(4 139 108 53 86)(5 140 109 54 87)(6 141 110 55 88)(7 142 111 56 89)(8 143 112 57 90)(9 144 113 58 91)(10 145 114 59 92)(11 146 115 60 93)(12 147 116 61 94)(13 148 117 62 95)(14 149 118 63 96)(15 150 119 64 65)(16 151 120 33 66)(17 152 121 34 67)(18 153 122 35 68)(19 154 123 36 69)(20 155 124 37 70)(21 156 125 38 71)(22 157 126 39 72)(23 158 127 40 73)(24 159 128 41 74)(25 160 97 42 75)(26 129 98 43 76)(27 130 99 44 77)(28 131 100 45 78)(29 132 101 46 79)(30 133 102 47 80)(31 134 103 48 81)(32 135 104 49 82)
(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)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)(33 49)(35 51)(37 53)(39 55)(41 57)(43 59)(45 61)(47 63)(66 82)(68 84)(70 86)(72 88)(74 90)(76 92)(78 94)(80 96)(98 114)(100 116)(102 118)(104 120)(106 122)(108 124)(110 126)(112 128)(129 145)(131 147)(133 149)(135 151)(137 153)(139 155)(141 157)(143 159)

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

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

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

200 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 16A ··· 16H 16I 16J 16K 16L 20A ··· 20H 20I 20J 20K 20L 32A ··· 32P 40A ··· 40P 40Q ··· 40X 80A ··· 80AF 80AG ··· 80AV 160A ··· 160BL order 1 2 2 4 4 4 5 5 5 5 8 8 8 8 8 8 10 10 10 10 10 10 10 10 16 ··· 16 16 16 16 16 20 ··· 20 20 20 20 20 32 ··· 32 40 ··· 40 40 ··· 40 80 ··· 80 80 ··· 80 160 ··· 160 size 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C5 C8 C8 C10 C10 C16 C16 C20 C20 C40 C40 C80 C80 M6(2) C5×M6(2) kernel C5×M6(2) C160 C2×C80 C80 C2×C40 M6(2) C40 C2×C20 C32 C2×C16 C20 C2×C10 C16 C2×C8 C8 C2×C4 C4 C22 C5 C1 # reps 1 2 1 2 2 4 4 4 8 4 8 8 8 8 16 16 32 32 8 32

Matrix representation of C5×M6(2) in GL3(𝔽641) generated by

 562 0 0 0 1 0 0 0 1
,
 1 0 0 0 392 639 0 603 249
,
 640 0 0 0 1 0 0 392 640
G:=sub<GL(3,GF(641))| [562,0,0,0,1,0,0,0,1],[1,0,0,0,392,603,0,639,249],[640,0,0,0,1,392,0,0,640] >;

C5×M6(2) in GAP, Magma, Sage, TeX

C_5\times M_6(2)
% in TeX

G:=Group("C5xM6(2)");
// GroupNames label

G:=SmallGroup(320,175);
// by ID

G=gap.SmallGroup(320,175);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,2269,80,102,124]);
// Polycyclic

G:=Group<a,b,c|a^5=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^17>;
// generators/relations

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