direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C42⋊2C2, C42⋊2C22, C4⋊C4⋊5C22, (C4×C44)⋊3C2, C22⋊C4.2C22, C23.3(C2×C22), C22.46(C4○D4), (C2×C44).68C22, (C2×C22).81C23, (C22×C22).3C22, C22.16(C22×C22), (C11×C4⋊C4)⋊14C2, (C2×C4).8(C2×C22), C2.9(C11×C4○D4), (C11×C22⋊C4).5C2, SmallGroup(352,161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C42⋊2C2
G = < a,b,c,d | a11=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >
Subgroups: 84 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C23, C11, C42, C22⋊C4, C4⋊C4, C22, C22, C42⋊2C2, C44, C2×C22, C2×C22, C2×C44, C22×C22, C4×C44, C11×C22⋊C4, C11×C4⋊C4, C11×C42⋊2C2
Quotients: C1, C2, C22, C23, C11, C4○D4, C22, C42⋊2C2, C2×C22, C22×C22, C11×C4○D4, C11×C42⋊2C2
►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)
(1 139 51 145)(2 140 52 146)(3 141 53 147)(4 142 54 148)(5 143 55 149)(6 133 45 150)(7 134 46 151)(8 135 47 152)(9 136 48 153)(10 137 49 154)(11 138 50 144)(12 110 38 93)(13 100 39 94)(14 101 40 95)(15 102 41 96)(16 103 42 97)(17 104 43 98)(18 105 44 99)(19 106 34 89)(20 107 35 90)(21 108 36 91)(22 109 37 92)(23 121 174 86)(24 111 175 87)(25 112 176 88)(26 113 166 78)(27 114 167 79)(28 115 168 80)(29 116 169 81)(30 117 170 82)(31 118 171 83)(32 119 172 84)(33 120 173 85)(56 122 69 157)(57 123 70 158)(58 124 71 159)(59 125 72 160)(60 126 73 161)(61 127 74 162)(62 128 75 163)(63 129 76 164)(64 130 77 165)(65 131 67 155)(66 132 68 156)
(1 79 70 101)(2 80 71 102)(3 81 72 103)(4 82 73 104)(5 83 74 105)(6 84 75 106)(7 85 76 107)(8 86 77 108)(9 87 67 109)(10 88 68 110)(11 78 69 100)(12 154 176 132)(13 144 166 122)(14 145 167 123)(15 146 168 124)(16 147 169 125)(17 148 170 126)(18 149 171 127)(19 150 172 128)(20 151 173 129)(21 152 174 130)(22 153 175 131)(23 165 36 135)(24 155 37 136)(25 156 38 137)(26 157 39 138)(27 158 40 139)(28 159 41 140)(29 160 42 141)(30 161 43 142)(31 162 44 143)(32 163 34 133)(33 164 35 134)(45 119 62 89)(46 120 63 90)(47 121 64 91)(48 111 65 92)(49 112 66 93)(50 113 56 94)(51 114 57 95)(52 115 58 96)(53 116 59 97)(54 117 60 98)(55 118 61 99)
(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 34)(20 35)(21 36)(22 37)(23 174)(24 175)(25 176)(26 166)(27 167)(28 168)(29 169)(30 170)(31 171)(32 172)(33 173)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 89)(85 90)(86 91)(87 92)(88 93)(100 113)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 121)(109 111)(110 112)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 153)(132 154)(133 163)(134 164)(135 165)(136 155)(137 156)(138 157)(139 158)(140 159)(141 160)(142 161)(143 162)
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), (1,139,51,145)(2,140,52,146)(3,141,53,147)(4,142,54,148)(5,143,55,149)(6,133,45,150)(7,134,46,151)(8,135,47,152)(9,136,48,153)(10,137,49,154)(11,138,50,144)(12,110,38,93)(13,100,39,94)(14,101,40,95)(15,102,41,96)(16,103,42,97)(17,104,43,98)(18,105,44,99)(19,106,34,89)(20,107,35,90)(21,108,36,91)(22,109,37,92)(23,121,174,86)(24,111,175,87)(25,112,176,88)(26,113,166,78)(27,114,167,79)(28,115,168,80)(29,116,169,81)(30,117,170,82)(31,118,171,83)(32,119,172,84)(33,120,173,85)(56,122,69,157)(57,123,70,158)(58,124,71,159)(59,125,72,160)(60,126,73,161)(61,127,74,162)(62,128,75,163)(63,129,76,164)(64,130,77,165)(65,131,67,155)(66,132,68,156), (1,79,70,101)(2,80,71,102)(3,81,72,103)(4,82,73,104)(5,83,74,105)(6,84,75,106)(7,85,76,107)(8,86,77,108)(9,87,67,109)(10,88,68,110)(11,78,69,100)(12,154,176,132)(13,144,166,122)(14,145,167,123)(15,146,168,124)(16,147,169,125)(17,148,170,126)(18,149,171,127)(19,150,172,128)(20,151,173,129)(21,152,174,130)(22,153,175,131)(23,165,36,135)(24,155,37,136)(25,156,38,137)(26,157,39,138)(27,158,40,139)(28,159,41,140)(29,160,42,141)(30,161,43,142)(31,162,44,143)(32,163,34,133)(33,164,35,134)(45,119,62,89)(46,120,63,90)(47,121,64,91)(48,111,65,92)(49,112,66,93)(50,113,56,94)(51,114,57,95)(52,115,58,96)(53,116,59,97)(54,117,60,98)(55,118,61,99), (12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,34)(20,35)(21,36)(22,37)(23,174)(24,175)(25,176)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,89)(85,90)(86,91)(87,92)(88,93)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121)(109,111)(110,112)(122,144)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,151)(130,152)(131,153)(132,154)(133,163)(134,164)(135,165)(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162)>;
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), (1,139,51,145)(2,140,52,146)(3,141,53,147)(4,142,54,148)(5,143,55,149)(6,133,45,150)(7,134,46,151)(8,135,47,152)(9,136,48,153)(10,137,49,154)(11,138,50,144)(12,110,38,93)(13,100,39,94)(14,101,40,95)(15,102,41,96)(16,103,42,97)(17,104,43,98)(18,105,44,99)(19,106,34,89)(20,107,35,90)(21,108,36,91)(22,109,37,92)(23,121,174,86)(24,111,175,87)(25,112,176,88)(26,113,166,78)(27,114,167,79)(28,115,168,80)(29,116,169,81)(30,117,170,82)(31,118,171,83)(32,119,172,84)(33,120,173,85)(56,122,69,157)(57,123,70,158)(58,124,71,159)(59,125,72,160)(60,126,73,161)(61,127,74,162)(62,128,75,163)(63,129,76,164)(64,130,77,165)(65,131,67,155)(66,132,68,156), (1,79,70,101)(2,80,71,102)(3,81,72,103)(4,82,73,104)(5,83,74,105)(6,84,75,106)(7,85,76,107)(8,86,77,108)(9,87,67,109)(10,88,68,110)(11,78,69,100)(12,154,176,132)(13,144,166,122)(14,145,167,123)(15,146,168,124)(16,147,169,125)(17,148,170,126)(18,149,171,127)(19,150,172,128)(20,151,173,129)(21,152,174,130)(22,153,175,131)(23,165,36,135)(24,155,37,136)(25,156,38,137)(26,157,39,138)(27,158,40,139)(28,159,41,140)(29,160,42,141)(30,161,43,142)(31,162,44,143)(32,163,34,133)(33,164,35,134)(45,119,62,89)(46,120,63,90)(47,121,64,91)(48,111,65,92)(49,112,66,93)(50,113,56,94)(51,114,57,95)(52,115,58,96)(53,116,59,97)(54,117,60,98)(55,118,61,99), (12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,34)(20,35)(21,36)(22,37)(23,174)(24,175)(25,176)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,89)(85,90)(86,91)(87,92)(88,93)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121)(109,111)(110,112)(122,144)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,151)(130,152)(131,153)(132,154)(133,163)(134,164)(135,165)(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162) );
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)], [(1,139,51,145),(2,140,52,146),(3,141,53,147),(4,142,54,148),(5,143,55,149),(6,133,45,150),(7,134,46,151),(8,135,47,152),(9,136,48,153),(10,137,49,154),(11,138,50,144),(12,110,38,93),(13,100,39,94),(14,101,40,95),(15,102,41,96),(16,103,42,97),(17,104,43,98),(18,105,44,99),(19,106,34,89),(20,107,35,90),(21,108,36,91),(22,109,37,92),(23,121,174,86),(24,111,175,87),(25,112,176,88),(26,113,166,78),(27,114,167,79),(28,115,168,80),(29,116,169,81),(30,117,170,82),(31,118,171,83),(32,119,172,84),(33,120,173,85),(56,122,69,157),(57,123,70,158),(58,124,71,159),(59,125,72,160),(60,126,73,161),(61,127,74,162),(62,128,75,163),(63,129,76,164),(64,130,77,165),(65,131,67,155),(66,132,68,156)], [(1,79,70,101),(2,80,71,102),(3,81,72,103),(4,82,73,104),(5,83,74,105),(6,84,75,106),(7,85,76,107),(8,86,77,108),(9,87,67,109),(10,88,68,110),(11,78,69,100),(12,154,176,132),(13,144,166,122),(14,145,167,123),(15,146,168,124),(16,147,169,125),(17,148,170,126),(18,149,171,127),(19,150,172,128),(20,151,173,129),(21,152,174,130),(22,153,175,131),(23,165,36,135),(24,155,37,136),(25,156,38,137),(26,157,39,138),(27,158,40,139),(28,159,41,140),(29,160,42,141),(30,161,43,142),(31,162,44,143),(32,163,34,133),(33,164,35,134),(45,119,62,89),(46,120,63,90),(47,121,64,91),(48,111,65,92),(49,112,66,93),(50,113,56,94),(51,114,57,95),(52,115,58,96),(53,116,59,97),(54,117,60,98),(55,118,61,99)], [(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,34),(20,35),(21,36),(22,37),(23,174),(24,175),(25,176),(26,166),(27,167),(28,168),(29,169),(30,170),(31,171),(32,172),(33,173),(78,94),(79,95),(80,96),(81,97),(82,98),(83,99),(84,89),(85,90),(86,91),(87,92),(88,93),(100,113),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,121),(109,111),(110,112),(122,144),(123,145),(124,146),(125,147),(126,148),(127,149),(128,150),(129,151),(130,152),(131,153),(132,154),(133,163),(134,164),(135,165),(136,155),(137,156),(138,157),(139,158),(140,159),(141,160),(142,161),(143,162)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22AN | 44A | ··· | 44BH | 44BI | ··· | 44CL |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | C4○D4 | C11×C4○D4 |
| kernel | C11×C42⋊2C2 | C4×C44 | C11×C22⋊C4 | C11×C4⋊C4 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C22 | C2 |
| # reps | 1 | 1 | 3 | 3 | 10 | 10 | 30 | 30 | 6 | 60 |
Matrix representation of C11×C42⋊2C2 ►in GL4(𝔽89) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 21 | 87 | 0 | 0 |
| 43 | 68 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 34 |
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 72 | 87 |
| 0 | 0 | 55 | 17 |
| 1 | 0 | 0 | 0 |
| 21 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 88 |
G:=sub<GL(4,GF(89))| [8,0,0,0,0,8,0,0,0,0,2,0,0,0,0,2],[21,43,0,0,87,68,0,0,0,0,34,0,0,0,0,34],[34,0,0,0,0,34,0,0,0,0,72,55,0,0,87,17],[1,21,0,0,0,88,0,0,0,0,1,72,0,0,0,88] >;
C11×C42⋊2C2 in GAP, Magma, Sage, TeX
C_{11}\times C_4^2\rtimes_2C_2 % in TeX
G:=Group("C11xC4^2:2C2"); // GroupNames label
G:=SmallGroup(352,161);
// by ID
G=gap.SmallGroup(352,161);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,1591,3242,410]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,d*c*d=b^2*c^-1>;
// generators/relations