direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×Q16, C8.C22, Q8.C22, C88.3C2, C22.16D4, C44.19C22, C4.3(C2×C22), C2.5(D4×C11), (Q8×C11).2C2, SmallGroup(176,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×Q16
G = < a,b,c | a11=b8=1, c2=b4, ab=ba, 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)
(1 167 51 30 73 40 66 21)(2 168 52 31 74 41 56 22)(3 169 53 32 75 42 57 12)(4 170 54 33 76 43 58 13)(5 171 55 23 77 44 59 14)(6 172 45 24 67 34 60 15)(7 173 46 25 68 35 61 16)(8 174 47 26 69 36 62 17)(9 175 48 27 70 37 63 18)(10 176 49 28 71 38 64 19)(11 166 50 29 72 39 65 20)(78 160 92 122 109 136 116 153)(79 161 93 123 110 137 117 154)(80 162 94 124 100 138 118 144)(81 163 95 125 101 139 119 145)(82 164 96 126 102 140 120 146)(83 165 97 127 103 141 121 147)(84 155 98 128 104 142 111 148)(85 156 99 129 105 143 112 149)(86 157 89 130 106 133 113 150)(87 158 90 131 107 134 114 151)(88 159 91 132 108 135 115 152)
(1 117 73 93)(2 118 74 94)(3 119 75 95)(4 120 76 96)(5 121 77 97)(6 111 67 98)(7 112 68 99)(8 113 69 89)(9 114 70 90)(10 115 71 91)(11 116 72 92)(12 145 32 125)(13 146 33 126)(14 147 23 127)(15 148 24 128)(16 149 25 129)(17 150 26 130)(18 151 27 131)(19 152 28 132)(20 153 29 122)(21 154 30 123)(22 144 31 124)(34 155 172 142)(35 156 173 143)(36 157 174 133)(37 158 175 134)(38 159 176 135)(39 160 166 136)(40 161 167 137)(41 162 168 138)(42 163 169 139)(43 164 170 140)(44 165 171 141)(45 104 60 84)(46 105 61 85)(47 106 62 86)(48 107 63 87)(49 108 64 88)(50 109 65 78)(51 110 66 79)(52 100 56 80)(53 101 57 81)(54 102 58 82)(55 103 59 83)
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), (1,167,51,30,73,40,66,21)(2,168,52,31,74,41,56,22)(3,169,53,32,75,42,57,12)(4,170,54,33,76,43,58,13)(5,171,55,23,77,44,59,14)(6,172,45,24,67,34,60,15)(7,173,46,25,68,35,61,16)(8,174,47,26,69,36,62,17)(9,175,48,27,70,37,63,18)(10,176,49,28,71,38,64,19)(11,166,50,29,72,39,65,20)(78,160,92,122,109,136,116,153)(79,161,93,123,110,137,117,154)(80,162,94,124,100,138,118,144)(81,163,95,125,101,139,119,145)(82,164,96,126,102,140,120,146)(83,165,97,127,103,141,121,147)(84,155,98,128,104,142,111,148)(85,156,99,129,105,143,112,149)(86,157,89,130,106,133,113,150)(87,158,90,131,107,134,114,151)(88,159,91,132,108,135,115,152), (1,117,73,93)(2,118,74,94)(3,119,75,95)(4,120,76,96)(5,121,77,97)(6,111,67,98)(7,112,68,99)(8,113,69,89)(9,114,70,90)(10,115,71,91)(11,116,72,92)(12,145,32,125)(13,146,33,126)(14,147,23,127)(15,148,24,128)(16,149,25,129)(17,150,26,130)(18,151,27,131)(19,152,28,132)(20,153,29,122)(21,154,30,123)(22,144,31,124)(34,155,172,142)(35,156,173,143)(36,157,174,133)(37,158,175,134)(38,159,176,135)(39,160,166,136)(40,161,167,137)(41,162,168,138)(42,163,169,139)(43,164,170,140)(44,165,171,141)(45,104,60,84)(46,105,61,85)(47,106,62,86)(48,107,63,87)(49,108,64,88)(50,109,65,78)(51,110,66,79)(52,100,56,80)(53,101,57,81)(54,102,58,82)(55,103,59,83)>;
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), (1,167,51,30,73,40,66,21)(2,168,52,31,74,41,56,22)(3,169,53,32,75,42,57,12)(4,170,54,33,76,43,58,13)(5,171,55,23,77,44,59,14)(6,172,45,24,67,34,60,15)(7,173,46,25,68,35,61,16)(8,174,47,26,69,36,62,17)(9,175,48,27,70,37,63,18)(10,176,49,28,71,38,64,19)(11,166,50,29,72,39,65,20)(78,160,92,122,109,136,116,153)(79,161,93,123,110,137,117,154)(80,162,94,124,100,138,118,144)(81,163,95,125,101,139,119,145)(82,164,96,126,102,140,120,146)(83,165,97,127,103,141,121,147)(84,155,98,128,104,142,111,148)(85,156,99,129,105,143,112,149)(86,157,89,130,106,133,113,150)(87,158,90,131,107,134,114,151)(88,159,91,132,108,135,115,152), (1,117,73,93)(2,118,74,94)(3,119,75,95)(4,120,76,96)(5,121,77,97)(6,111,67,98)(7,112,68,99)(8,113,69,89)(9,114,70,90)(10,115,71,91)(11,116,72,92)(12,145,32,125)(13,146,33,126)(14,147,23,127)(15,148,24,128)(16,149,25,129)(17,150,26,130)(18,151,27,131)(19,152,28,132)(20,153,29,122)(21,154,30,123)(22,144,31,124)(34,155,172,142)(35,156,173,143)(36,157,174,133)(37,158,175,134)(38,159,176,135)(39,160,166,136)(40,161,167,137)(41,162,168,138)(42,163,169,139)(43,164,170,140)(44,165,171,141)(45,104,60,84)(46,105,61,85)(47,106,62,86)(48,107,63,87)(49,108,64,88)(50,109,65,78)(51,110,66,79)(52,100,56,80)(53,101,57,81)(54,102,58,82)(55,103,59,83) );
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)], [(1,167,51,30,73,40,66,21),(2,168,52,31,74,41,56,22),(3,169,53,32,75,42,57,12),(4,170,54,33,76,43,58,13),(5,171,55,23,77,44,59,14),(6,172,45,24,67,34,60,15),(7,173,46,25,68,35,61,16),(8,174,47,26,69,36,62,17),(9,175,48,27,70,37,63,18),(10,176,49,28,71,38,64,19),(11,166,50,29,72,39,65,20),(78,160,92,122,109,136,116,153),(79,161,93,123,110,137,117,154),(80,162,94,124,100,138,118,144),(81,163,95,125,101,139,119,145),(82,164,96,126,102,140,120,146),(83,165,97,127,103,141,121,147),(84,155,98,128,104,142,111,148),(85,156,99,129,105,143,112,149),(86,157,89,130,106,133,113,150),(87,158,90,131,107,134,114,151),(88,159,91,132,108,135,115,152)], [(1,117,73,93),(2,118,74,94),(3,119,75,95),(4,120,76,96),(5,121,77,97),(6,111,67,98),(7,112,68,99),(8,113,69,89),(9,114,70,90),(10,115,71,91),(11,116,72,92),(12,145,32,125),(13,146,33,126),(14,147,23,127),(15,148,24,128),(16,149,25,129),(17,150,26,130),(18,151,27,131),(19,152,28,132),(20,153,29,122),(21,154,30,123),(22,144,31,124),(34,155,172,142),(35,156,173,143),(36,157,174,133),(37,158,175,134),(38,159,176,135),(39,160,166,136),(40,161,167,137),(41,162,168,138),(42,163,169,139),(43,164,170,140),(44,165,171,141),(45,104,60,84),(46,105,61,85),(47,106,62,86),(48,107,63,87),(49,108,64,88),(50,109,65,78),(51,110,66,79),(52,100,56,80),(53,101,57,81),(54,102,58,82),(55,103,59,83)]])
C11×Q16 is a maximal subgroup of
C8.6D22 C11⋊Q32 Q16⋊D11 D88⋊5C2
77 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 11A | ··· | 11J | 22A | ··· | 22J | 44A | ··· | 44J | 44K | ··· | 44AD | 88A | ··· | 88T |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D4 | Q16 | D4×C11 | C11×Q16 |
| kernel | C11×Q16 | C88 | Q8×C11 | Q16 | C8 | Q8 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 1 | 2 | 10 | 20 |
Matrix representation of C11×Q16 ►in GL2(𝔽23) generated by
| 3 | 0 |
| 0 | 3 |
| 0 | 16 |
| 10 | 5 |
| 13 | 19 |
| 8 | 10 |
G:=sub<GL(2,GF(23))| [3,0,0,3],[0,10,16,5],[13,8,19,10] >;
C11×Q16 in GAP, Magma, Sage, TeX
C_{11}\times Q_{16} % in TeX
G:=Group("C11xQ16"); // GroupNames label
G:=SmallGroup(176,26);
// by ID
G=gap.SmallGroup(176,26);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-2,440,461,446,2643,1328,58]);
// Polycyclic
G:=Group<a,b,c|a^11=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export