direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C11×SD32, D8.C22, C16⋊2C22, C176⋊6C2, Q16⋊1C22, C22.16D8, C44.37D4, C88.25C22, C8.3(C2×C22), C4.2(D4×C11), C2.4(C11×D8), (C11×Q16)⋊5C2, (C11×D8).2C2, SmallGroup(352,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×SD32
G = < a,b,c | a11=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Smallest permutation representation of C11×SD32
►On 176 points
Generators in S176
(1 146 125 140 161 90 30 59 98 44 66)(2 147 126 141 162 91 31 60 99 45 67)(3 148 127 142 163 92 32 61 100 46 68)(4 149 128 143 164 93 17 62 101 47 69)(5 150 113 144 165 94 18 63 102 48 70)(6 151 114 129 166 95 19 64 103 33 71)(7 152 115 130 167 96 20 49 104 34 72)(8 153 116 131 168 81 21 50 105 35 73)(9 154 117 132 169 82 22 51 106 36 74)(10 155 118 133 170 83 23 52 107 37 75)(11 156 119 134 171 84 24 53 108 38 76)(12 157 120 135 172 85 25 54 109 39 77)(13 158 121 136 173 86 26 55 110 40 78)(14 159 122 137 174 87 27 56 111 41 79)(15 160 123 138 175 88 28 57 112 42 80)(16 145 124 139 176 89 29 58 97 43 65)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 19)(18 26)(20 24)(21 31)(23 29)(25 27)(28 32)(33 47)(34 38)(35 45)(37 43)(39 41)(40 48)(42 46)(49 53)(50 60)(52 58)(54 56)(55 63)(57 61)(62 64)(65 75)(67 73)(68 80)(69 71)(70 78)(72 76)(77 79)(81 91)(83 89)(84 96)(85 87)(86 94)(88 92)(93 95)(97 107)(99 105)(100 112)(101 103)(102 110)(104 108)(109 111)(113 121)(114 128)(115 119)(116 126)(118 124)(120 122)(123 127)(129 143)(130 134)(131 141)(133 139)(135 137)(136 144)(138 142)(145 155)(147 153)(148 160)(149 151)(150 158)(152 156)(157 159)(162 168)(163 175)(164 166)(165 173)(167 171)(170 176)(172 174)
G:=sub<Sym(176)| (1,146,125,140,161,90,30,59,98,44,66)(2,147,126,141,162,91,31,60,99,45,67)(3,148,127,142,163,92,32,61,100,46,68)(4,149,128,143,164,93,17,62,101,47,69)(5,150,113,144,165,94,18,63,102,48,70)(6,151,114,129,166,95,19,64,103,33,71)(7,152,115,130,167,96,20,49,104,34,72)(8,153,116,131,168,81,21,50,105,35,73)(9,154,117,132,169,82,22,51,106,36,74)(10,155,118,133,170,83,23,52,107,37,75)(11,156,119,134,171,84,24,53,108,38,76)(12,157,120,135,172,85,25,54,109,39,77)(13,158,121,136,173,86,26,55,110,40,78)(14,159,122,137,174,87,27,56,111,41,79)(15,160,123,138,175,88,28,57,112,42,80)(16,145,124,139,176,89,29,58,97,43,65), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,47)(34,38)(35,45)(37,43)(39,41)(40,48)(42,46)(49,53)(50,60)(52,58)(54,56)(55,63)(57,61)(62,64)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(97,107)(99,105)(100,112)(101,103)(102,110)(104,108)(109,111)(113,121)(114,128)(115,119)(116,126)(118,124)(120,122)(123,127)(129,143)(130,134)(131,141)(133,139)(135,137)(136,144)(138,142)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159)(162,168)(163,175)(164,166)(165,173)(167,171)(170,176)(172,174)>;
G:=Group( (1,146,125,140,161,90,30,59,98,44,66)(2,147,126,141,162,91,31,60,99,45,67)(3,148,127,142,163,92,32,61,100,46,68)(4,149,128,143,164,93,17,62,101,47,69)(5,150,113,144,165,94,18,63,102,48,70)(6,151,114,129,166,95,19,64,103,33,71)(7,152,115,130,167,96,20,49,104,34,72)(8,153,116,131,168,81,21,50,105,35,73)(9,154,117,132,169,82,22,51,106,36,74)(10,155,118,133,170,83,23,52,107,37,75)(11,156,119,134,171,84,24,53,108,38,76)(12,157,120,135,172,85,25,54,109,39,77)(13,158,121,136,173,86,26,55,110,40,78)(14,159,122,137,174,87,27,56,111,41,79)(15,160,123,138,175,88,28,57,112,42,80)(16,145,124,139,176,89,29,58,97,43,65), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,47)(34,38)(35,45)(37,43)(39,41)(40,48)(42,46)(49,53)(50,60)(52,58)(54,56)(55,63)(57,61)(62,64)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)(81,91)(83,89)(84,96)(85,87)(86,94)(88,92)(93,95)(97,107)(99,105)(100,112)(101,103)(102,110)(104,108)(109,111)(113,121)(114,128)(115,119)(116,126)(118,124)(120,122)(123,127)(129,143)(130,134)(131,141)(133,139)(135,137)(136,144)(138,142)(145,155)(147,153)(148,160)(149,151)(150,158)(152,156)(157,159)(162,168)(163,175)(164,166)(165,173)(167,171)(170,176)(172,174) );
G=PermutationGroup([[(1,146,125,140,161,90,30,59,98,44,66),(2,147,126,141,162,91,31,60,99,45,67),(3,148,127,142,163,92,32,61,100,46,68),(4,149,128,143,164,93,17,62,101,47,69),(5,150,113,144,165,94,18,63,102,48,70),(6,151,114,129,166,95,19,64,103,33,71),(7,152,115,130,167,96,20,49,104,34,72),(8,153,116,131,168,81,21,50,105,35,73),(9,154,117,132,169,82,22,51,106,36,74),(10,155,118,133,170,83,23,52,107,37,75),(11,156,119,134,171,84,24,53,108,38,76),(12,157,120,135,172,85,25,54,109,39,77),(13,158,121,136,173,86,26,55,110,40,78),(14,159,122,137,174,87,27,56,111,41,79),(15,160,123,138,175,88,28,57,112,42,80),(16,145,124,139,176,89,29,58,97,43,65)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,19),(18,26),(20,24),(21,31),(23,29),(25,27),(28,32),(33,47),(34,38),(35,45),(37,43),(39,41),(40,48),(42,46),(49,53),(50,60),(52,58),(54,56),(55,63),(57,61),(62,64),(65,75),(67,73),(68,80),(69,71),(70,78),(72,76),(77,79),(81,91),(83,89),(84,96),(85,87),(86,94),(88,92),(93,95),(97,107),(99,105),(100,112),(101,103),(102,110),(104,108),(109,111),(113,121),(114,128),(115,119),(116,126),(118,124),(120,122),(123,127),(129,143),(130,134),(131,141),(133,139),(135,137),(136,144),(138,142),(145,155),(147,153),(148,160),(149,151),(150,158),(152,156),(157,159),(162,168),(163,175),(164,166),(165,173),(167,171),(170,176),(172,174)]])
121 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 11A | ··· | 11J | 16A | 16B | 16C | 16D | 22A | ··· | 22J | 22K | ··· | 22T | 44A | ··· | 44J | 44K | ··· | 44T | 88A | ··· | 88T | 176A | ··· | 176AN |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 16 | 16 | 16 | 16 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 8 | 2 | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
121 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | D4 | D8 | SD32 | D4×C11 | C11×D8 | C11×SD32 |
| kernel | C11×SD32 | C176 | C11×D8 | C11×Q16 | SD32 | C16 | D8 | Q16 | C44 | C22 | C11 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 2 | 4 | 10 | 20 | 40 |
Matrix representation of C11×SD32 ►in GL2(𝔽23) generated by
| 9 | 0 |
| 0 | 9 |
| 16 | 15 |
| 20 | 0 |
| 22 | 10 |
| 0 | 1 |
G:=sub<GL(2,GF(23))| [9,0,0,9],[16,20,15,0],[22,0,10,1] >;
C11×SD32 in GAP, Magma, Sage, TeX
C_{11}\times {\rm SD}_{32} % in TeX
G:=Group("C11xSD32"); // GroupNames label
G:=SmallGroup(352,61);
// by ID
G=gap.SmallGroup(352,61);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,1056,553,3171,1593,165,7924,3970,88]);
// 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^7>;
// generators/relations
Export