metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C11⋊2Q16, Q8.D11, C4.4D22, C22.10D4, C44.4C22, Dic22.2C2, C11⋊C8.C2, (Q8×C11).1C2, C2.7(C11⋊D4), SmallGroup(176,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11⋊Q16
G = < a,b,c | a11=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11⋊Q16
►Regular action 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)
(1 155 32 144 21 166 43 133)(2 165 33 154 22 176 44 143)(3 164 23 153 12 175 34 142)(4 163 24 152 13 174 35 141)(5 162 25 151 14 173 36 140)(6 161 26 150 15 172 37 139)(7 160 27 149 16 171 38 138)(8 159 28 148 17 170 39 137)(9 158 29 147 18 169 40 136)(10 157 30 146 19 168 41 135)(11 156 31 145 20 167 42 134)(45 120 78 98 56 131 67 109)(46 119 79 97 57 130 68 108)(47 118 80 96 58 129 69 107)(48 117 81 95 59 128 70 106)(49 116 82 94 60 127 71 105)(50 115 83 93 61 126 72 104)(51 114 84 92 62 125 73 103)(52 113 85 91 63 124 74 102)(53 112 86 90 64 123 75 101)(54 111 87 89 65 122 76 100)(55 121 88 99 66 132 77 110)
(1 65 21 54)(2 66 22 55)(3 56 12 45)(4 57 13 46)(5 58 14 47)(6 59 15 48)(7 60 16 49)(8 61 17 50)(9 62 18 51)(10 63 19 52)(11 64 20 53)(23 78 34 67)(24 79 35 68)(25 80 36 69)(26 81 37 70)(27 82 38 71)(28 83 39 72)(29 84 40 73)(30 85 41 74)(31 86 42 75)(32 87 43 76)(33 88 44 77)(89 166 100 155)(90 167 101 156)(91 168 102 157)(92 169 103 158)(93 170 104 159)(94 171 105 160)(95 172 106 161)(96 173 107 162)(97 174 108 163)(98 175 109 164)(99 176 110 165)(111 133 122 144)(112 134 123 145)(113 135 124 146)(114 136 125 147)(115 137 126 148)(116 138 127 149)(117 139 128 150)(118 140 129 151)(119 141 130 152)(120 142 131 153)(121 143 132 154)
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), (1,155,32,144,21,166,43,133)(2,165,33,154,22,176,44,143)(3,164,23,153,12,175,34,142)(4,163,24,152,13,174,35,141)(5,162,25,151,14,173,36,140)(6,161,26,150,15,172,37,139)(7,160,27,149,16,171,38,138)(8,159,28,148,17,170,39,137)(9,158,29,147,18,169,40,136)(10,157,30,146,19,168,41,135)(11,156,31,145,20,167,42,134)(45,120,78,98,56,131,67,109)(46,119,79,97,57,130,68,108)(47,118,80,96,58,129,69,107)(48,117,81,95,59,128,70,106)(49,116,82,94,60,127,71,105)(50,115,83,93,61,126,72,104)(51,114,84,92,62,125,73,103)(52,113,85,91,63,124,74,102)(53,112,86,90,64,123,75,101)(54,111,87,89,65,122,76,100)(55,121,88,99,66,132,77,110), (1,65,21,54)(2,66,22,55)(3,56,12,45)(4,57,13,46)(5,58,14,47)(6,59,15,48)(7,60,16,49)(8,61,17,50)(9,62,18,51)(10,63,19,52)(11,64,20,53)(23,78,34,67)(24,79,35,68)(25,80,36,69)(26,81,37,70)(27,82,38,71)(28,83,39,72)(29,84,40,73)(30,85,41,74)(31,86,42,75)(32,87,43,76)(33,88,44,77)(89,166,100,155)(90,167,101,156)(91,168,102,157)(92,169,103,158)(93,170,104,159)(94,171,105,160)(95,172,106,161)(96,173,107,162)(97,174,108,163)(98,175,109,164)(99,176,110,165)(111,133,122,144)(112,134,123,145)(113,135,124,146)(114,136,125,147)(115,137,126,148)(116,138,127,149)(117,139,128,150)(118,140,129,151)(119,141,130,152)(120,142,131,153)(121,143,132,154)>;
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), (1,155,32,144,21,166,43,133)(2,165,33,154,22,176,44,143)(3,164,23,153,12,175,34,142)(4,163,24,152,13,174,35,141)(5,162,25,151,14,173,36,140)(6,161,26,150,15,172,37,139)(7,160,27,149,16,171,38,138)(8,159,28,148,17,170,39,137)(9,158,29,147,18,169,40,136)(10,157,30,146,19,168,41,135)(11,156,31,145,20,167,42,134)(45,120,78,98,56,131,67,109)(46,119,79,97,57,130,68,108)(47,118,80,96,58,129,69,107)(48,117,81,95,59,128,70,106)(49,116,82,94,60,127,71,105)(50,115,83,93,61,126,72,104)(51,114,84,92,62,125,73,103)(52,113,85,91,63,124,74,102)(53,112,86,90,64,123,75,101)(54,111,87,89,65,122,76,100)(55,121,88,99,66,132,77,110), (1,65,21,54)(2,66,22,55)(3,56,12,45)(4,57,13,46)(5,58,14,47)(6,59,15,48)(7,60,16,49)(8,61,17,50)(9,62,18,51)(10,63,19,52)(11,64,20,53)(23,78,34,67)(24,79,35,68)(25,80,36,69)(26,81,37,70)(27,82,38,71)(28,83,39,72)(29,84,40,73)(30,85,41,74)(31,86,42,75)(32,87,43,76)(33,88,44,77)(89,166,100,155)(90,167,101,156)(91,168,102,157)(92,169,103,158)(93,170,104,159)(94,171,105,160)(95,172,106,161)(96,173,107,162)(97,174,108,163)(98,175,109,164)(99,176,110,165)(111,133,122,144)(112,134,123,145)(113,135,124,146)(114,136,125,147)(115,137,126,148)(116,138,127,149)(117,139,128,150)(118,140,129,151)(119,141,130,152)(120,142,131,153)(121,143,132,154) );
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)], [(1,155,32,144,21,166,43,133),(2,165,33,154,22,176,44,143),(3,164,23,153,12,175,34,142),(4,163,24,152,13,174,35,141),(5,162,25,151,14,173,36,140),(6,161,26,150,15,172,37,139),(7,160,27,149,16,171,38,138),(8,159,28,148,17,170,39,137),(9,158,29,147,18,169,40,136),(10,157,30,146,19,168,41,135),(11,156,31,145,20,167,42,134),(45,120,78,98,56,131,67,109),(46,119,79,97,57,130,68,108),(47,118,80,96,58,129,69,107),(48,117,81,95,59,128,70,106),(49,116,82,94,60,127,71,105),(50,115,83,93,61,126,72,104),(51,114,84,92,62,125,73,103),(52,113,85,91,63,124,74,102),(53,112,86,90,64,123,75,101),(54,111,87,89,65,122,76,100),(55,121,88,99,66,132,77,110)], [(1,65,21,54),(2,66,22,55),(3,56,12,45),(4,57,13,46),(5,58,14,47),(6,59,15,48),(7,60,16,49),(8,61,17,50),(9,62,18,51),(10,63,19,52),(11,64,20,53),(23,78,34,67),(24,79,35,68),(25,80,36,69),(26,81,37,70),(27,82,38,71),(28,83,39,72),(29,84,40,73),(30,85,41,74),(31,86,42,75),(32,87,43,76),(33,88,44,77),(89,166,100,155),(90,167,101,156),(91,168,102,157),(92,169,103,158),(93,170,104,159),(94,171,105,160),(95,172,106,161),(96,173,107,162),(97,174,108,163),(98,175,109,164),(99,176,110,165),(111,133,122,144),(112,134,123,145),(113,135,124,146),(114,136,125,147),(115,137,126,148),(116,138,127,149),(117,139,128,150),(118,140,129,151),(119,141,130,152),(120,142,131,153),(121,143,132,154)]])
C11⋊Q16 is a maximal subgroup of
D4.D22 Q8.D22 Q16×D11 Q16⋊D11 C44.C23 D4.8D22 D4.9D22
C11⋊Q16 is a maximal quotient of C44.Q8 C22.Q16 Q8⋊Dic11
32 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44O |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 4 | 44 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | Q16 | D11 | D22 | C11⋊D4 | C11⋊Q16 |
| kernel | C11⋊Q16 | C11⋊C8 | Dic22 | Q8×C11 | C22 | C11 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 5 | 10 | 5 |
Matrix representation of C11⋊Q16 ►in GL4(𝔽89) generated by
| 0 | 1 | 0 | 0 |
| 88 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 29 | 0 | 0 |
| 40 | 49 | 0 | 0 |
| 0 | 0 | 32 | 57 |
| 0 | 0 | 32 | 32 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 3 | 41 |
| 0 | 0 | 41 | 86 |
G:=sub<GL(4,GF(89))| [0,88,0,0,1,47,0,0,0,0,1,0,0,0,0,1],[40,40,0,0,29,49,0,0,0,0,32,32,0,0,57,32],[88,0,0,0,0,88,0,0,0,0,3,41,0,0,41,86] >;
C11⋊Q16 in GAP, Magma, Sage, TeX
C_{11}\rtimes Q_{16} % in TeX
G:=Group("C11:Q16"); // GroupNames label
G:=SmallGroup(176,17);
// by ID
G=gap.SmallGroup(176,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,40,61,46,182,97,42,4004]);
// Polycyclic
G:=Group<a,b,c|a^11=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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