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## G = C11×M5(2)  order 352 = 25·11

### Direct product of C11 and M5(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×M5(2)
 Chief series C1 — C2 — C4 — C8 — C88 — C176 — C11×M5(2)
 Lower central C1 — C2 — C11×M5(2)
 Upper central C1 — C88 — C11×M5(2)

Generators and relations for C11×M5(2)
G = < a,b,c | a11=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Smallest permutation representation of C11×M5(2)
On 176 points
Generators in S176
(1 133 23 175 151 119 37 92 59 102 74)(2 134 24 176 152 120 38 93 60 103 75)(3 135 25 161 153 121 39 94 61 104 76)(4 136 26 162 154 122 40 95 62 105 77)(5 137 27 163 155 123 41 96 63 106 78)(6 138 28 164 156 124 42 81 64 107 79)(7 139 29 165 157 125 43 82 49 108 80)(8 140 30 166 158 126 44 83 50 109 65)(9 141 31 167 159 127 45 84 51 110 66)(10 142 32 168 160 128 46 85 52 111 67)(11 143 17 169 145 113 47 86 53 112 68)(12 144 18 170 146 114 48 87 54 97 69)(13 129 19 171 147 115 33 88 55 98 70)(14 130 20 172 148 116 34 89 56 99 71)(15 131 21 173 149 117 35 90 57 100 72)(16 132 22 174 150 118 36 91 58 101 73)
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(97 105)(99 107)(101 109)(103 111)(114 122)(116 124)(118 126)(120 128)(130 138)(132 140)(134 142)(136 144)(146 154)(148 156)(150 158)(152 160)(162 170)(164 172)(166 174)(168 176)

G:=sub<Sym(176)| (1,133,23,175,151,119,37,92,59,102,74)(2,134,24,176,152,120,38,93,60,103,75)(3,135,25,161,153,121,39,94,61,104,76)(4,136,26,162,154,122,40,95,62,105,77)(5,137,27,163,155,123,41,96,63,106,78)(6,138,28,164,156,124,42,81,64,107,79)(7,139,29,165,157,125,43,82,49,108,80)(8,140,30,166,158,126,44,83,50,109,65)(9,141,31,167,159,127,45,84,51,110,66)(10,142,32,168,160,128,46,85,52,111,67)(11,143,17,169,145,113,47,86,53,112,68)(12,144,18,170,146,114,48,87,54,97,69)(13,129,19,171,147,115,33,88,55,98,70)(14,130,20,172,148,116,34,89,56,99,71)(15,131,21,173,149,117,35,90,57,100,72)(16,132,22,174,150,118,36,91,58,101,73), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111)(114,122)(116,124)(118,126)(120,128)(130,138)(132,140)(134,142)(136,144)(146,154)(148,156)(150,158)(152,160)(162,170)(164,172)(166,174)(168,176)>;

G:=Group( (1,133,23,175,151,119,37,92,59,102,74)(2,134,24,176,152,120,38,93,60,103,75)(3,135,25,161,153,121,39,94,61,104,76)(4,136,26,162,154,122,40,95,62,105,77)(5,137,27,163,155,123,41,96,63,106,78)(6,138,28,164,156,124,42,81,64,107,79)(7,139,29,165,157,125,43,82,49,108,80)(8,140,30,166,158,126,44,83,50,109,65)(9,141,31,167,159,127,45,84,51,110,66)(10,142,32,168,160,128,46,85,52,111,67)(11,143,17,169,145,113,47,86,53,112,68)(12,144,18,170,146,114,48,87,54,97,69)(13,129,19,171,147,115,33,88,55,98,70)(14,130,20,172,148,116,34,89,56,99,71)(15,131,21,173,149,117,35,90,57,100,72)(16,132,22,174,150,118,36,91,58,101,73), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111)(114,122)(116,124)(118,126)(120,128)(130,138)(132,140)(134,142)(136,144)(146,154)(148,156)(150,158)(152,160)(162,170)(164,172)(166,174)(168,176) );

G=PermutationGroup([(1,133,23,175,151,119,37,92,59,102,74),(2,134,24,176,152,120,38,93,60,103,75),(3,135,25,161,153,121,39,94,61,104,76),(4,136,26,162,154,122,40,95,62,105,77),(5,137,27,163,155,123,41,96,63,106,78),(6,138,28,164,156,124,42,81,64,107,79),(7,139,29,165,157,125,43,82,49,108,80),(8,140,30,166,158,126,44,83,50,109,65),(9,141,31,167,159,127,45,84,51,110,66),(10,142,32,168,160,128,46,85,52,111,67),(11,143,17,169,145,113,47,86,53,112,68),(12,144,18,170,146,114,48,87,54,97,69),(13,129,19,171,147,115,33,88,55,98,70),(14,130,20,172,148,116,34,89,56,99,71),(15,131,21,173,149,117,35,90,57,100,72),(16,132,22,174,150,118,36,91,58,101,73)], [(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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(34,42),(36,44),(38,46),(40,48),(50,58),(52,60),(54,62),(56,64),(65,73),(67,75),(69,77),(71,79),(81,89),(83,91),(85,93),(87,95),(97,105),(99,107),(101,109),(103,111),(114,122),(116,124),(118,126),(120,128),(130,138),(132,140),(134,142),(136,144),(146,154),(148,156),(150,158),(152,160),(162,170),(164,172),(166,174),(168,176)])

220 conjugacy classes

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 8E 8F 11A ··· 11J 16A ··· 16H 22A ··· 22J 22K ··· 22T 44A ··· 44T 44U ··· 44AD 88A ··· 88AN 88AO ··· 88BH 176A ··· 176CB order 1 2 2 4 4 4 8 8 8 8 8 8 11 ··· 11 16 ··· 16 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 88 ··· 88 176 ··· 176 size 1 1 2 1 1 2 1 1 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

220 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C8 C8 C11 C22 C22 C44 C44 C88 C88 M5(2) C11×M5(2) kernel C11×M5(2) C176 C2×C88 C88 C2×C44 C44 C2×C22 M5(2) C16 C2×C8 C8 C2×C4 C4 C22 C11 C1 # reps 1 2 1 2 2 4 4 10 20 10 20 20 40 40 4 40

Matrix representation of C11×M5(2) in GL2(𝔽353) generated by

 337 0 0 337
,
 220 351 231 133
,
 1 0 220 352
G:=sub<GL(2,GF(353))| [337,0,0,337],[220,231,351,133],[1,220,0,352] >;

C11×M5(2) in GAP, Magma, Sage, TeX

C_{11}\times M_5(2)
% in TeX

G:=Group("C11xM5(2)");
// GroupNames label

G:=SmallGroup(352,59);
// by ID

G=gap.SmallGroup(352,59);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,264,2137,69,88]);
// Polycyclic

G:=Group<a,b,c|a^11=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations

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