direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×C8.C4, C88.5C4, C8.1C44, C44.68D4, M4(2).2C22, C4.8(C2×C44), (C2×C8).5C22, (C2×C22).2Q8, (C2×C88).15C2, C44.45(C2×C4), C22.(Q8×C11), C4.19(D4×C11), C22.14(C4⋊C4), (C2×C44).119C22, (C11×M4(2)).4C2, C2.5(C11×C4⋊C4), (C2×C4).22(C2×C22), SmallGroup(352,57)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C8.C4
G = < a,b,c | a11=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C11×C8.C4
►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)
(1 108 71 115 46 90 66 79)(2 109 72 116 47 91 56 80)(3 110 73 117 48 92 57 81)(4 100 74 118 49 93 58 82)(5 101 75 119 50 94 59 83)(6 102 76 120 51 95 60 84)(7 103 77 121 52 96 61 85)(8 104 67 111 53 97 62 86)(9 105 68 112 54 98 63 87)(10 106 69 113 55 99 64 88)(11 107 70 114 45 89 65 78)(12 139 34 164 31 146 172 128)(13 140 35 165 32 147 173 129)(14 141 36 155 33 148 174 130)(15 142 37 156 23 149 175 131)(16 143 38 157 24 150 176 132)(17 133 39 158 25 151 166 122)(18 134 40 159 26 152 167 123)(19 135 41 160 27 153 168 124)(20 136 42 161 28 154 169 125)(21 137 43 162 29 144 170 126)(22 138 44 163 30 145 171 127)
(1 159 71 134 46 123 66 152)(2 160 72 135 47 124 56 153)(3 161 73 136 48 125 57 154)(4 162 74 137 49 126 58 144)(5 163 75 138 50 127 59 145)(6 164 76 139 51 128 60 146)(7 165 77 140 52 129 61 147)(8 155 67 141 53 130 62 148)(9 156 68 142 54 131 63 149)(10 157 69 143 55 132 64 150)(11 158 70 133 45 122 65 151)(12 95 172 84 31 102 34 120)(13 96 173 85 32 103 35 121)(14 97 174 86 33 104 36 111)(15 98 175 87 23 105 37 112)(16 99 176 88 24 106 38 113)(17 89 166 78 25 107 39 114)(18 90 167 79 26 108 40 115)(19 91 168 80 27 109 41 116)(20 92 169 81 28 110 42 117)(21 93 170 82 29 100 43 118)(22 94 171 83 30 101 44 119)
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), (1,108,71,115,46,90,66,79)(2,109,72,116,47,91,56,80)(3,110,73,117,48,92,57,81)(4,100,74,118,49,93,58,82)(5,101,75,119,50,94,59,83)(6,102,76,120,51,95,60,84)(7,103,77,121,52,96,61,85)(8,104,67,111,53,97,62,86)(9,105,68,112,54,98,63,87)(10,106,69,113,55,99,64,88)(11,107,70,114,45,89,65,78)(12,139,34,164,31,146,172,128)(13,140,35,165,32,147,173,129)(14,141,36,155,33,148,174,130)(15,142,37,156,23,149,175,131)(16,143,38,157,24,150,176,132)(17,133,39,158,25,151,166,122)(18,134,40,159,26,152,167,123)(19,135,41,160,27,153,168,124)(20,136,42,161,28,154,169,125)(21,137,43,162,29,144,170,126)(22,138,44,163,30,145,171,127), (1,159,71,134,46,123,66,152)(2,160,72,135,47,124,56,153)(3,161,73,136,48,125,57,154)(4,162,74,137,49,126,58,144)(5,163,75,138,50,127,59,145)(6,164,76,139,51,128,60,146)(7,165,77,140,52,129,61,147)(8,155,67,141,53,130,62,148)(9,156,68,142,54,131,63,149)(10,157,69,143,55,132,64,150)(11,158,70,133,45,122,65,151)(12,95,172,84,31,102,34,120)(13,96,173,85,32,103,35,121)(14,97,174,86,33,104,36,111)(15,98,175,87,23,105,37,112)(16,99,176,88,24,106,38,113)(17,89,166,78,25,107,39,114)(18,90,167,79,26,108,40,115)(19,91,168,80,27,109,41,116)(20,92,169,81,28,110,42,117)(21,93,170,82,29,100,43,118)(22,94,171,83,30,101,44,119)>;
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), (1,108,71,115,46,90,66,79)(2,109,72,116,47,91,56,80)(3,110,73,117,48,92,57,81)(4,100,74,118,49,93,58,82)(5,101,75,119,50,94,59,83)(6,102,76,120,51,95,60,84)(7,103,77,121,52,96,61,85)(8,104,67,111,53,97,62,86)(9,105,68,112,54,98,63,87)(10,106,69,113,55,99,64,88)(11,107,70,114,45,89,65,78)(12,139,34,164,31,146,172,128)(13,140,35,165,32,147,173,129)(14,141,36,155,33,148,174,130)(15,142,37,156,23,149,175,131)(16,143,38,157,24,150,176,132)(17,133,39,158,25,151,166,122)(18,134,40,159,26,152,167,123)(19,135,41,160,27,153,168,124)(20,136,42,161,28,154,169,125)(21,137,43,162,29,144,170,126)(22,138,44,163,30,145,171,127), (1,159,71,134,46,123,66,152)(2,160,72,135,47,124,56,153)(3,161,73,136,48,125,57,154)(4,162,74,137,49,126,58,144)(5,163,75,138,50,127,59,145)(6,164,76,139,51,128,60,146)(7,165,77,140,52,129,61,147)(8,155,67,141,53,130,62,148)(9,156,68,142,54,131,63,149)(10,157,69,143,55,132,64,150)(11,158,70,133,45,122,65,151)(12,95,172,84,31,102,34,120)(13,96,173,85,32,103,35,121)(14,97,174,86,33,104,36,111)(15,98,175,87,23,105,37,112)(16,99,176,88,24,106,38,113)(17,89,166,78,25,107,39,114)(18,90,167,79,26,108,40,115)(19,91,168,80,27,109,41,116)(20,92,169,81,28,110,42,117)(21,93,170,82,29,100,43,118)(22,94,171,83,30,101,44,119) );
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)], [(1,108,71,115,46,90,66,79),(2,109,72,116,47,91,56,80),(3,110,73,117,48,92,57,81),(4,100,74,118,49,93,58,82),(5,101,75,119,50,94,59,83),(6,102,76,120,51,95,60,84),(7,103,77,121,52,96,61,85),(8,104,67,111,53,97,62,86),(9,105,68,112,54,98,63,87),(10,106,69,113,55,99,64,88),(11,107,70,114,45,89,65,78),(12,139,34,164,31,146,172,128),(13,140,35,165,32,147,173,129),(14,141,36,155,33,148,174,130),(15,142,37,156,23,149,175,131),(16,143,38,157,24,150,176,132),(17,133,39,158,25,151,166,122),(18,134,40,159,26,152,167,123),(19,135,41,160,27,153,168,124),(20,136,42,161,28,154,169,125),(21,137,43,162,29,144,170,126),(22,138,44,163,30,145,171,127)], [(1,159,71,134,46,123,66,152),(2,160,72,135,47,124,56,153),(3,161,73,136,48,125,57,154),(4,162,74,137,49,126,58,144),(5,163,75,138,50,127,59,145),(6,164,76,139,51,128,60,146),(7,165,77,140,52,129,61,147),(8,155,67,141,53,130,62,148),(9,156,68,142,54,131,63,149),(10,157,69,143,55,132,64,150),(11,158,70,133,45,122,65,151),(12,95,172,84,31,102,34,120),(13,96,173,85,32,103,35,121),(14,97,174,86,33,104,36,111),(15,98,175,87,23,105,37,112),(16,99,176,88,24,106,38,113),(17,89,166,78,25,107,39,114),(18,90,167,79,26,108,40,115),(19,91,168,80,27,109,41,116),(20,92,169,81,28,110,42,117),(21,93,170,82,29,100,43,118),(22,94,171,83,30,101,44,119)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22T | 44A | ··· | 44T | 44U | ··· | 44AD | 88A | ··· | 88AN | 88AO | ··· | 88CB |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C4 | C11 | C22 | C22 | C44 | D4 | Q8 | C8.C4 | D4×C11 | Q8×C11 | C11×C8.C4 |
| kernel | C11×C8.C4 | C2×C88 | C11×M4(2) | C88 | C8.C4 | C2×C8 | M4(2) | C8 | C44 | C2×C22 | C11 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 40 | 1 | 1 | 4 | 10 | 10 | 40 |
Matrix representation of C11×C8.C4 ►in GL2(𝔽89) generated by
| 67 | 0 |
| 0 | 67 |
| 12 | 0 |
| 79 | 52 |
| 22 | 1 |
| 16 | 67 |
G:=sub<GL(2,GF(89))| [67,0,0,67],[12,79,0,52],[22,16,1,67] >;
C11×C8.C4 in GAP, Magma, Sage, TeX
C_{11}\times C_8.C_4 % in TeX
G:=Group("C11xC8.C4"); // GroupNames label
G:=SmallGroup(352,57);
// by ID
G=gap.SmallGroup(352,57);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,271,5283,117,88]);
// Polycyclic
G:=Group<a,b,c|a^11=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export