direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×M5(2), C4.C88, C176⋊7C2, C16⋊3C22, C88.6C4, C44.4C8, C8.2C44, C22.C88, C88.29C22, C2.3(C2×C88), (C2×C4).5C44, (C2×C22).1C8, (C2×C8).8C22, C8.8(C2×C22), (C2×C88).18C2, C4.12(C2×C44), (C2×C44).14C4, C22.13(C2×C8), C44.49(C2×C4), SmallGroup(352,59)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×M5(2)
G = < a,b,c | a11=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C11×M5(2)
►On 176 points
Generators in S176
(1 104 160 49 116 144 37 87 80 169 31)(2 105 145 50 117 129 38 88 65 170 32)(3 106 146 51 118 130 39 89 66 171 17)(4 107 147 52 119 131 40 90 67 172 18)(5 108 148 53 120 132 41 91 68 173 19)(6 109 149 54 121 133 42 92 69 174 20)(7 110 150 55 122 134 43 93 70 175 21)(8 111 151 56 123 135 44 94 71 176 22)(9 112 152 57 124 136 45 95 72 161 23)(10 97 153 58 125 137 46 96 73 162 24)(11 98 154 59 126 138 47 81 74 163 25)(12 99 155 60 127 139 48 82 75 164 26)(13 100 156 61 128 140 33 83 76 165 27)(14 101 157 62 113 141 34 84 77 166 28)(15 102 158 63 114 142 35 85 78 167 29)(16 103 159 64 115 143 36 86 79 168 30)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)(113 121)(115 123)(117 125)(119 127)(129 137)(131 139)(133 141)(135 143)(145 153)(147 155)(149 157)(151 159)(162 170)(164 172)(166 174)(168 176)
G:=sub<Sym(176)| (1,104,160,49,116,144,37,87,80,169,31)(2,105,145,50,117,129,38,88,65,170,32)(3,106,146,51,118,130,39,89,66,171,17)(4,107,147,52,119,131,40,90,67,172,18)(5,108,148,53,120,132,41,91,68,173,19)(6,109,149,54,121,133,42,92,69,174,20)(7,110,150,55,122,134,43,93,70,175,21)(8,111,151,56,123,135,44,94,71,176,22)(9,112,152,57,124,136,45,95,72,161,23)(10,97,153,58,125,137,46,96,73,162,24)(11,98,154,59,126,138,47,81,74,163,25)(12,99,155,60,127,139,48,82,75,164,26)(13,100,156,61,128,140,33,83,76,165,27)(14,101,157,62,113,141,34,84,77,166,28)(15,102,158,63,114,142,35,85,78,167,29)(16,103,159,64,115,143,36,86,79,168,30), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111)(113,121)(115,123)(117,125)(119,127)(129,137)(131,139)(133,141)(135,143)(145,153)(147,155)(149,157)(151,159)(162,170)(164,172)(166,174)(168,176)>;
G:=Group( (1,104,160,49,116,144,37,87,80,169,31)(2,105,145,50,117,129,38,88,65,170,32)(3,106,146,51,118,130,39,89,66,171,17)(4,107,147,52,119,131,40,90,67,172,18)(5,108,148,53,120,132,41,91,68,173,19)(6,109,149,54,121,133,42,92,69,174,20)(7,110,150,55,122,134,43,93,70,175,21)(8,111,151,56,123,135,44,94,71,176,22)(9,112,152,57,124,136,45,95,72,161,23)(10,97,153,58,125,137,46,96,73,162,24)(11,98,154,59,126,138,47,81,74,163,25)(12,99,155,60,127,139,48,82,75,164,26)(13,100,156,61,128,140,33,83,76,165,27)(14,101,157,62,113,141,34,84,77,166,28)(15,102,158,63,114,142,35,85,78,167,29)(16,103,159,64,115,143,36,86,79,168,30), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111)(113,121)(115,123)(117,125)(119,127)(129,137)(131,139)(133,141)(135,143)(145,153)(147,155)(149,157)(151,159)(162,170)(164,172)(166,174)(168,176) );
G=PermutationGroup([[(1,104,160,49,116,144,37,87,80,169,31),(2,105,145,50,117,129,38,88,65,170,32),(3,106,146,51,118,130,39,89,66,171,17),(4,107,147,52,119,131,40,90,67,172,18),(5,108,148,53,120,132,41,91,68,173,19),(6,109,149,54,121,133,42,92,69,174,20),(7,110,150,55,122,134,43,93,70,175,21),(8,111,151,56,123,135,44,94,71,176,22),(9,112,152,57,124,136,45,95,72,161,23),(10,97,153,58,125,137,46,96,73,162,24),(11,98,154,59,126,138,47,81,74,163,25),(12,99,155,60,127,139,48,82,75,164,26),(13,100,156,61,128,140,33,83,76,165,27),(14,101,157,62,113,141,34,84,77,166,28),(15,102,158,63,114,142,35,85,78,167,29),(16,103,159,64,115,143,36,86,79,168,30)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(34,42),(36,44),(38,46),(40,48),(50,58),(52,60),(54,62),(56,64),(65,73),(67,75),(69,77),(71,79),(82,90),(84,92),(86,94),(88,96),(97,105),(99,107),(101,109),(103,111),(113,121),(115,123),(117,125),(119,127),(129,137),(131,139),(133,141),(135,143),(145,153),(147,155),(149,157),(151,159),(162,170),(164,172),(166,174),(168,176)]])
220 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 11A | ··· | 11J | 16A | ··· | 16H | 22A | ··· | 22J | 22K | ··· | 22T | 44A | ··· | 44T | 44U | ··· | 44AD | 88A | ··· | 88AN | 88AO | ··· | 88BH | 176A | ··· | 176CB |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 16 | ··· | 16 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
220 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C11 | C22 | C22 | C44 | C44 | C88 | C88 | M5(2) | C11×M5(2) |
| kernel | C11×M5(2) | C176 | C2×C88 | C88 | C2×C44 | C44 | C2×C22 | M5(2) | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C11 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 10 | 20 | 10 | 20 | 20 | 40 | 40 | 4 | 40 |
Matrix representation of C11×M5(2) ►in GL2(𝔽353) generated by
| 337 | 0 |
| 0 | 337 |
| 220 | 351 |
| 231 | 133 |
| 1 | 0 |
| 220 | 352 |
G:=sub<GL(2,GF(353))| [337,0,0,337],[220,231,351,133],[1,220,0,352] >;
C11×M5(2) in GAP, Magma, Sage, TeX
C_{11}\times M_5(2) % in TeX
G:=Group("C11xM5(2)"); // GroupNames label
G:=SmallGroup(352,59);
// by ID
G=gap.SmallGroup(352,59);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,264,2137,69,88]);
// 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^9>;
// generators/relations
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