metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11⋊C16, C22.C8, C44.2C4, C88.2C2, C8.2D11, C4.2Dic11, C2.(C11⋊C8), SmallGroup(176,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C11⋊C16 |
Generators and relations for C11⋊C16
G = < a,b | a11=b16=1, bab-1=a-1 >
Smallest permutation representation of C11⋊C16
►Regular action on 176 points
Generators in S176
(1 81 151 98 46 138 61 24 115 174 76)(2 77 175 116 25 62 139 47 99 152 82)(3 83 153 100 48 140 63 26 117 176 78)(4 79 161 118 27 64 141 33 101 154 84)(5 85 155 102 34 142 49 28 119 162 80)(6 65 163 120 29 50 143 35 103 156 86)(7 87 157 104 36 144 51 30 121 164 66)(8 67 165 122 31 52 129 37 105 158 88)(9 89 159 106 38 130 53 32 123 166 68)(10 69 167 124 17 54 131 39 107 160 90)(11 91 145 108 40 132 55 18 125 168 70)(12 71 169 126 19 56 133 41 109 146 92)(13 93 147 110 42 134 57 20 127 170 72)(14 73 171 128 21 58 135 43 111 148 94)(15 95 149 112 44 136 59 22 113 172 74)(16 75 173 114 23 60 137 45 97 150 96)
(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)
G:=sub<Sym(176)| (1,81,151,98,46,138,61,24,115,174,76)(2,77,175,116,25,62,139,47,99,152,82)(3,83,153,100,48,140,63,26,117,176,78)(4,79,161,118,27,64,141,33,101,154,84)(5,85,155,102,34,142,49,28,119,162,80)(6,65,163,120,29,50,143,35,103,156,86)(7,87,157,104,36,144,51,30,121,164,66)(8,67,165,122,31,52,129,37,105,158,88)(9,89,159,106,38,130,53,32,123,166,68)(10,69,167,124,17,54,131,39,107,160,90)(11,91,145,108,40,132,55,18,125,168,70)(12,71,169,126,19,56,133,41,109,146,92)(13,93,147,110,42,134,57,20,127,170,72)(14,73,171,128,21,58,135,43,111,148,94)(15,95,149,112,44,136,59,22,113,172,74)(16,75,173,114,23,60,137,45,97,150,96), (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)>;
G:=Group( (1,81,151,98,46,138,61,24,115,174,76)(2,77,175,116,25,62,139,47,99,152,82)(3,83,153,100,48,140,63,26,117,176,78)(4,79,161,118,27,64,141,33,101,154,84)(5,85,155,102,34,142,49,28,119,162,80)(6,65,163,120,29,50,143,35,103,156,86)(7,87,157,104,36,144,51,30,121,164,66)(8,67,165,122,31,52,129,37,105,158,88)(9,89,159,106,38,130,53,32,123,166,68)(10,69,167,124,17,54,131,39,107,160,90)(11,91,145,108,40,132,55,18,125,168,70)(12,71,169,126,19,56,133,41,109,146,92)(13,93,147,110,42,134,57,20,127,170,72)(14,73,171,128,21,58,135,43,111,148,94)(15,95,149,112,44,136,59,22,113,172,74)(16,75,173,114,23,60,137,45,97,150,96), (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) );
G=PermutationGroup([[(1,81,151,98,46,138,61,24,115,174,76),(2,77,175,116,25,62,139,47,99,152,82),(3,83,153,100,48,140,63,26,117,176,78),(4,79,161,118,27,64,141,33,101,154,84),(5,85,155,102,34,142,49,28,119,162,80),(6,65,163,120,29,50,143,35,103,156,86),(7,87,157,104,36,144,51,30,121,164,66),(8,67,165,122,31,52,129,37,105,158,88),(9,89,159,106,38,130,53,32,123,166,68),(10,69,167,124,17,54,131,39,107,160,90),(11,91,145,108,40,132,55,18,125,168,70),(12,71,169,126,19,56,133,41,109,146,92),(13,93,147,110,42,134,57,20,127,170,72),(14,73,171,128,21,58,135,43,111,148,94),(15,95,149,112,44,136,59,22,113,172,74),(16,75,173,114,23,60,137,45,97,150,96)], [(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)]])
C11⋊C16 is a maximal subgroup of
C16×D11 D22.C8 C44.C8 C11⋊D16 D8.D11 C8.6D22 C11⋊Q32
C11⋊C16 is a maximal quotient of C11⋊C32
56 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 16A | ··· | 16H | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 16 | ··· | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D11 | Dic11 | C11⋊C8 | C11⋊C16 |
| kernel | C11⋊C16 | C88 | C44 | C22 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 5 | 5 | 10 | 20 |
Matrix representation of C11⋊C16 ►in GL3(𝔽353) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 352 | 95 |
| 293 | 0 | 0 |
| 0 | 16 | 339 |
| 0 | 94 | 337 |
G:=sub<GL(3,GF(353))| [1,0,0,0,0,352,0,1,95],[293,0,0,0,16,94,0,339,337] >;
C11⋊C16 in GAP, Magma, Sage, TeX
C_{11}\rtimes C_{16} % in TeX
G:=Group("C11:C16"); // GroupNames label
G:=SmallGroup(176,1);
// by ID
G=gap.SmallGroup(176,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,10,26,42,4004]);
// Polycyclic
G:=Group<a,b|a^11=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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