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## G = C11×C8.C4order 352 = 25·11

### Direct product of C11 and C8.C4

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C11×C8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C44 — C11×M4(2) — C11×C8.C4
 Lower central C1 — C2 — C4 — C11×C8.C4
 Upper central C1 — C44 — C2×C44 — C11×C8.C4

Generators and relations for C11×C8.C4
G = < a,b,c | a11=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C11×C8.C4
On 176 points
Generators in S176
(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)
(1 117 59 103 74 90 55 79)(2 118 60 104 75 91 45 80)(3 119 61 105 76 92 46 81)(4 120 62 106 77 93 47 82)(5 121 63 107 67 94 48 83)(6 111 64 108 68 95 49 84)(7 112 65 109 69 96 50 85)(8 113 66 110 70 97 51 86)(9 114 56 100 71 98 52 87)(10 115 57 101 72 99 53 88)(11 116 58 102 73 89 54 78)(12 159 176 132 25 143 38 145)(13 160 166 122 26 133 39 146)(14 161 167 123 27 134 40 147)(15 162 168 124 28 135 41 148)(16 163 169 125 29 136 42 149)(17 164 170 126 30 137 43 150)(18 165 171 127 31 138 44 151)(19 155 172 128 32 139 34 152)(20 156 173 129 33 140 35 153)(21 157 174 130 23 141 36 154)(22 158 175 131 24 142 37 144)
(1 147 59 134 74 123 55 161)(2 148 60 135 75 124 45 162)(3 149 61 136 76 125 46 163)(4 150 62 137 77 126 47 164)(5 151 63 138 67 127 48 165)(6 152 64 139 68 128 49 155)(7 153 65 140 69 129 50 156)(8 154 66 141 70 130 51 157)(9 144 56 142 71 131 52 158)(10 145 57 143 72 132 53 159)(11 146 58 133 73 122 54 160)(12 115 38 101 25 99 176 88)(13 116 39 102 26 89 166 78)(14 117 40 103 27 90 167 79)(15 118 41 104 28 91 168 80)(16 119 42 105 29 92 169 81)(17 120 43 106 30 93 170 82)(18 121 44 107 31 94 171 83)(19 111 34 108 32 95 172 84)(20 112 35 109 33 96 173 85)(21 113 36 110 23 97 174 86)(22 114 37 100 24 98 175 87)

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

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

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

154 conjugacy classes

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

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C11 C22 C22 C44 D4 Q8 C8.C4 D4×C11 Q8×C11 C11×C8.C4 kernel C11×C8.C4 C2×C88 C11×M4(2) C88 C8.C4 C2×C8 M4(2) C8 C44 C2×C22 C11 C4 C22 C1 # reps 1 1 2 4 10 10 20 40 1 1 4 10 10 40

Matrix representation of C11×C8.C4 in GL2(𝔽89) generated by

 67 0 0 67
,
 12 0 79 52
,
 22 1 16 67
G:=sub<GL(2,GF(89))| [67,0,0,67],[12,79,0,52],[22,16,1,67] >;

C11×C8.C4 in GAP, Magma, Sage, TeX

C_{11}\times C_8.C_4
% in TeX

G:=Group("C11xC8.C4");
// GroupNames label

G:=SmallGroup(352,57);
// by ID

G=gap.SmallGroup(352,57);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,271,5283,117,88]);
// Polycyclic

G:=Group<a,b,c|a^11=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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