direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C22⋊C8, C22⋊C88, C44.65D4, C23.2C44, C22.8M4(2), (C2×C88)⋊3C2, (C2×C8)⋊1C22, (C2×C22)⋊1C8, C2.1(C2×C88), (C2×C4).3C44, (C2×C44).6C4, C22.11(C2×C8), C4.16(D4×C11), C22.9(C2×C44), (C22×C22).3C4, (C22×C4).2C22, (C22×C44).3C2, C2.2(C11×M4(2)), C22.20(C22⋊C4), (C2×C44).135C22, (C2×C22).38(C2×C4), (C2×C4).31(C2×C22), C2.2(C11×C22⋊C4), SmallGroup(352,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C22⋊C8
G = < a,b,c,d | a11=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C11×C22⋊C8
►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)
(12 74)(13 75)(14 76)(15 77)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 56)(31 57)(32 58)(33 59)(34 175)(35 176)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 78)(53 79)(54 80)(55 81)
(1 158)(2 159)(3 160)(4 161)(5 162)(6 163)(7 164)(8 165)(9 155)(10 156)(11 157)(12 74)(13 75)(14 76)(15 77)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 56)(31 57)(32 58)(33 59)(34 175)(35 176)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 78)(53 79)(54 80)(55 81)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 122)(96 123)(97 124)(98 125)(99 126)(100 134)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 141)(108 142)(109 143)(110 133)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 144)(119 145)(120 146)(121 147)
(1 54 114 70 125 166 108 26)(2 55 115 71 126 167 109 27)(3 45 116 72 127 168 110 28)(4 46 117 73 128 169 100 29)(5 47 118 74 129 170 101 30)(6 48 119 75 130 171 102 31)(7 49 120 76 131 172 103 32)(8 50 121 77 132 173 104 33)(9 51 111 67 122 174 105 23)(10 52 112 68 123 175 106 24)(11 53 113 69 124 176 107 25)(12 91 40 135 56 162 84 144)(13 92 41 136 57 163 85 145)(14 93 42 137 58 164 86 146)(15 94 43 138 59 165 87 147)(16 95 44 139 60 155 88 148)(17 96 34 140 61 156 78 149)(18 97 35 141 62 157 79 150)(19 98 36 142 63 158 80 151)(20 99 37 143 64 159 81 152)(21 89 38 133 65 160 82 153)(22 90 39 134 66 161 83 154)
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), (12,74)(13,75)(14,76)(15,77)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,56)(31,57)(32,58)(33,59)(34,175)(35,176)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,78)(53,79)(54,80)(55,81), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,155)(10,156)(11,157)(12,74)(13,75)(14,76)(15,77)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,56)(31,57)(32,58)(33,59)(34,175)(35,176)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,78)(53,79)(54,80)(55,81)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,122)(96,123)(97,124)(98,125)(99,126)(100,134)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,141)(108,142)(109,143)(110,133)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,144)(119,145)(120,146)(121,147), (1,54,114,70,125,166,108,26)(2,55,115,71,126,167,109,27)(3,45,116,72,127,168,110,28)(4,46,117,73,128,169,100,29)(5,47,118,74,129,170,101,30)(6,48,119,75,130,171,102,31)(7,49,120,76,131,172,103,32)(8,50,121,77,132,173,104,33)(9,51,111,67,122,174,105,23)(10,52,112,68,123,175,106,24)(11,53,113,69,124,176,107,25)(12,91,40,135,56,162,84,144)(13,92,41,136,57,163,85,145)(14,93,42,137,58,164,86,146)(15,94,43,138,59,165,87,147)(16,95,44,139,60,155,88,148)(17,96,34,140,61,156,78,149)(18,97,35,141,62,157,79,150)(19,98,36,142,63,158,80,151)(20,99,37,143,64,159,81,152)(21,89,38,133,65,160,82,153)(22,90,39,134,66,161,83,154)>;
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), (12,74)(13,75)(14,76)(15,77)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,56)(31,57)(32,58)(33,59)(34,175)(35,176)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,78)(53,79)(54,80)(55,81), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,155)(10,156)(11,157)(12,74)(13,75)(14,76)(15,77)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,56)(31,57)(32,58)(33,59)(34,175)(35,176)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,78)(53,79)(54,80)(55,81)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,122)(96,123)(97,124)(98,125)(99,126)(100,134)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,141)(108,142)(109,143)(110,133)(111,148)(112,149)(113,150)(114,151)(115,152)(116,153)(117,154)(118,144)(119,145)(120,146)(121,147), (1,54,114,70,125,166,108,26)(2,55,115,71,126,167,109,27)(3,45,116,72,127,168,110,28)(4,46,117,73,128,169,100,29)(5,47,118,74,129,170,101,30)(6,48,119,75,130,171,102,31)(7,49,120,76,131,172,103,32)(8,50,121,77,132,173,104,33)(9,51,111,67,122,174,105,23)(10,52,112,68,123,175,106,24)(11,53,113,69,124,176,107,25)(12,91,40,135,56,162,84,144)(13,92,41,136,57,163,85,145)(14,93,42,137,58,164,86,146)(15,94,43,138,59,165,87,147)(16,95,44,139,60,155,88,148)(17,96,34,140,61,156,78,149)(18,97,35,141,62,157,79,150)(19,98,36,142,63,158,80,151)(20,99,37,143,64,159,81,152)(21,89,38,133,65,160,82,153)(22,90,39,134,66,161,83,154) );
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)], [(12,74),(13,75),(14,76),(15,77),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,56),(31,57),(32,58),(33,59),(34,175),(35,176),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(51,88),(52,78),(53,79),(54,80),(55,81)], [(1,158),(2,159),(3,160),(4,161),(5,162),(6,163),(7,164),(8,165),(9,155),(10,156),(11,157),(12,74),(13,75),(14,76),(15,77),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,56),(31,57),(32,58),(33,59),(34,175),(35,176),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(51,88),(52,78),(53,79),(54,80),(55,81),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,122),(96,123),(97,124),(98,125),(99,126),(100,134),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,141),(108,142),(109,143),(110,133),(111,148),(112,149),(113,150),(114,151),(115,152),(116,153),(117,154),(118,144),(119,145),(120,146),(121,147)], [(1,54,114,70,125,166,108,26),(2,55,115,71,126,167,109,27),(3,45,116,72,127,168,110,28),(4,46,117,73,128,169,100,29),(5,47,118,74,129,170,101,30),(6,48,119,75,130,171,102,31),(7,49,120,76,131,172,103,32),(8,50,121,77,132,173,104,33),(9,51,111,67,122,174,105,23),(10,52,112,68,123,175,106,24),(11,53,113,69,124,176,107,25),(12,91,40,135,56,162,84,144),(13,92,41,136,57,163,85,145),(14,93,42,137,58,164,86,146),(15,94,43,138,59,165,87,147),(16,95,44,139,60,155,88,148),(17,96,34,140,61,156,78,149),(18,97,35,141,62,157,79,150),(19,98,36,142,63,158,80,151),(20,99,37,143,64,159,81,152),(21,89,38,133,65,160,82,153),(22,90,39,134,66,161,83,154)]])
220 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22AX | 44A | ··· | 44AN | 44AO | ··· | 44BH | 88A | ··· | 88CB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C11 | C22 | C22 | C44 | C44 | C88 | D4 | M4(2) | D4×C11 | C11×M4(2) |
| kernel | C11×C22⋊C8 | C2×C88 | C22×C44 | C2×C44 | C22×C22 | C2×C22 | C22⋊C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C44 | C22 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 10 | 20 | 10 | 20 | 20 | 80 | 2 | 2 | 20 | 20 |
Matrix representation of C11×C22⋊C8 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 88 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 56 | 88 |
| 1 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 12 | 0 | 0 |
| 0 | 56 | 87 |
| 0 | 38 | 33 |
G:=sub<GL(3,GF(89))| [1,0,0,0,32,0,0,0,32],[88,0,0,0,1,56,0,0,88],[1,0,0,0,88,0,0,0,88],[12,0,0,0,56,38,0,87,33] >;
C11×C22⋊C8 in GAP, Magma, Sage, TeX
C_{11}\times C_2^2\rtimes C_8 % in TeX
G:=Group("C11xC2^2:C8"); // GroupNames label
G:=SmallGroup(352,47);
// by ID
G=gap.SmallGroup(352,47);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,88]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
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