direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C42⋊C2, C42⋊1C22, C4⋊C4⋊6C22, (C2×C4)⋊4C44, (C4×C44)⋊2C2, (C2×C44)⋊9C4, C4.9(C2×C44), C44.46(C2×C4), C22⋊C4.3C22, C2.3(C22×C44), C23.6(C2×C22), (C22×C4).4C22, C22.5(C2×C44), C22.38(C4○D4), (C22×C44).14C2, (C2×C44).79C22, (C2×C22).72C23, C22.31(C22×C4), C22.6(C22×C22), (C22×C22).25C22, (C11×C4⋊C4)⋊15C2, C2.1(C11×C4○D4), (C2×C4).14(C2×C22), (C2×C22).22(C2×C4), (C11×C22⋊C4).6C2, SmallGroup(352,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C42⋊C2
G = < a,b,c,d | a11=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C11, C42, C22⋊C4, C4⋊C4, C22×C4, C22, C22, C22, C42⋊C2, C44, C44, C2×C22, C2×C22, C2×C22, C2×C44, C2×C44, C22×C22, C4×C44, C11×C22⋊C4, C11×C4⋊C4, C22×C44, C11×C42⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C11, C22×C4, C4○D4, C22, C42⋊C2, C44, C2×C22, C2×C44, C22×C22, C22×C44, C11×C4○D4, C11×C42⋊C2
►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)
(1 134 46 149)(2 135 47 150)(3 136 48 151)(4 137 49 152)(5 138 50 153)(6 139 51 154)(7 140 52 144)(8 141 53 145)(9 142 54 146)(10 143 55 147)(11 133 45 148)(12 98 27 102)(13 99 28 103)(14 89 29 104)(15 90 30 105)(16 91 31 106)(17 92 32 107)(18 93 33 108)(19 94 23 109)(20 95 24 110)(21 96 25 100)(22 97 26 101)(34 113 172 84)(35 114 173 85)(36 115 174 86)(37 116 175 87)(38 117 176 88)(39 118 166 78)(40 119 167 79)(41 120 168 80)(42 121 169 81)(43 111 170 82)(44 112 171 83)(56 156 68 127)(57 157 69 128)(58 158 70 129)(59 159 71 130)(60 160 72 131)(61 161 73 132)(62 162 74 122)(63 163 75 123)(64 164 76 124)(65 165 77 125)(66 155 67 126)
(1 79 63 105)(2 80 64 106)(3 81 65 107)(4 82 66 108)(5 83 56 109)(6 84 57 110)(7 85 58 100)(8 86 59 101)(9 87 60 102)(10 88 61 103)(11 78 62 104)(12 142 37 160)(13 143 38 161)(14 133 39 162)(15 134 40 163)(16 135 41 164)(17 136 42 165)(18 137 43 155)(19 138 44 156)(20 139 34 157)(21 140 35 158)(22 141 36 159)(23 153 171 127)(24 154 172 128)(25 144 173 129)(26 145 174 130)(27 146 175 131)(28 147 176 132)(29 148 166 122)(30 149 167 123)(31 150 168 124)(32 151 169 125)(33 152 170 126)(45 118 74 89)(46 119 75 90)(47 120 76 91)(48 121 77 92)(49 111 67 93)(50 112 68 94)(51 113 69 95)(52 114 70 96)(53 115 71 97)(54 116 72 98)(55 117 73 99)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 45)(12 175)(13 176)(14 166)(15 167)(16 168)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 44)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 67)(78 118)(79 119)(80 120)(81 121)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 100)(97 101)(98 102)(99 103)(122 133)(123 134)(124 135)(125 136)(126 137)(127 138)(128 139)(129 140)(130 141)(131 142)(132 143)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 155)(153 156)(154 157)
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), (1,134,46,149)(2,135,47,150)(3,136,48,151)(4,137,49,152)(5,138,50,153)(6,139,51,154)(7,140,52,144)(8,141,53,145)(9,142,54,146)(10,143,55,147)(11,133,45,148)(12,98,27,102)(13,99,28,103)(14,89,29,104)(15,90,30,105)(16,91,31,106)(17,92,32,107)(18,93,33,108)(19,94,23,109)(20,95,24,110)(21,96,25,100)(22,97,26,101)(34,113,172,84)(35,114,173,85)(36,115,174,86)(37,116,175,87)(38,117,176,88)(39,118,166,78)(40,119,167,79)(41,120,168,80)(42,121,169,81)(43,111,170,82)(44,112,171,83)(56,156,68,127)(57,157,69,128)(58,158,70,129)(59,159,71,130)(60,160,72,131)(61,161,73,132)(62,162,74,122)(63,163,75,123)(64,164,76,124)(65,165,77,125)(66,155,67,126), (1,79,63,105)(2,80,64,106)(3,81,65,107)(4,82,66,108)(5,83,56,109)(6,84,57,110)(7,85,58,100)(8,86,59,101)(9,87,60,102)(10,88,61,103)(11,78,62,104)(12,142,37,160)(13,143,38,161)(14,133,39,162)(15,134,40,163)(16,135,41,164)(17,136,42,165)(18,137,43,155)(19,138,44,156)(20,139,34,157)(21,140,35,158)(22,141,36,159)(23,153,171,127)(24,154,172,128)(25,144,173,129)(26,145,174,130)(27,146,175,131)(28,147,176,132)(29,148,166,122)(30,149,167,123)(31,150,168,124)(32,151,169,125)(33,152,170,126)(45,118,74,89)(46,119,75,90)(47,120,76,91)(48,121,77,92)(49,111,67,93)(50,112,68,94)(51,113,69,95)(52,114,70,96)(53,115,71,97)(54,116,72,98)(55,117,73,99), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,175)(13,176)(14,166)(15,167)(16,168)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,44)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,67)(78,118)(79,119)(80,120)(81,121)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,100)(97,101)(98,102)(99,103)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139)(129,140)(130,141)(131,142)(132,143)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,155)(153,156)(154,157)>;
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), (1,134,46,149)(2,135,47,150)(3,136,48,151)(4,137,49,152)(5,138,50,153)(6,139,51,154)(7,140,52,144)(8,141,53,145)(9,142,54,146)(10,143,55,147)(11,133,45,148)(12,98,27,102)(13,99,28,103)(14,89,29,104)(15,90,30,105)(16,91,31,106)(17,92,32,107)(18,93,33,108)(19,94,23,109)(20,95,24,110)(21,96,25,100)(22,97,26,101)(34,113,172,84)(35,114,173,85)(36,115,174,86)(37,116,175,87)(38,117,176,88)(39,118,166,78)(40,119,167,79)(41,120,168,80)(42,121,169,81)(43,111,170,82)(44,112,171,83)(56,156,68,127)(57,157,69,128)(58,158,70,129)(59,159,71,130)(60,160,72,131)(61,161,73,132)(62,162,74,122)(63,163,75,123)(64,164,76,124)(65,165,77,125)(66,155,67,126), (1,79,63,105)(2,80,64,106)(3,81,65,107)(4,82,66,108)(5,83,56,109)(6,84,57,110)(7,85,58,100)(8,86,59,101)(9,87,60,102)(10,88,61,103)(11,78,62,104)(12,142,37,160)(13,143,38,161)(14,133,39,162)(15,134,40,163)(16,135,41,164)(17,136,42,165)(18,137,43,155)(19,138,44,156)(20,139,34,157)(21,140,35,158)(22,141,36,159)(23,153,171,127)(24,154,172,128)(25,144,173,129)(26,145,174,130)(27,146,175,131)(28,147,176,132)(29,148,166,122)(30,149,167,123)(31,150,168,124)(32,151,169,125)(33,152,170,126)(45,118,74,89)(46,119,75,90)(47,120,76,91)(48,121,77,92)(49,111,67,93)(50,112,68,94)(51,113,69,95)(52,114,70,96)(53,115,71,97)(54,116,72,98)(55,117,73,99), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,175)(13,176)(14,166)(15,167)(16,168)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,44)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)(66,67)(78,118)(79,119)(80,120)(81,121)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,100)(97,101)(98,102)(99,103)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139)(129,140)(130,141)(131,142)(132,143)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,155)(153,156)(154,157) );
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)], [(1,134,46,149),(2,135,47,150),(3,136,48,151),(4,137,49,152),(5,138,50,153),(6,139,51,154),(7,140,52,144),(8,141,53,145),(9,142,54,146),(10,143,55,147),(11,133,45,148),(12,98,27,102),(13,99,28,103),(14,89,29,104),(15,90,30,105),(16,91,31,106),(17,92,32,107),(18,93,33,108),(19,94,23,109),(20,95,24,110),(21,96,25,100),(22,97,26,101),(34,113,172,84),(35,114,173,85),(36,115,174,86),(37,116,175,87),(38,117,176,88),(39,118,166,78),(40,119,167,79),(41,120,168,80),(42,121,169,81),(43,111,170,82),(44,112,171,83),(56,156,68,127),(57,157,69,128),(58,158,70,129),(59,159,71,130),(60,160,72,131),(61,161,73,132),(62,162,74,122),(63,163,75,123),(64,164,76,124),(65,165,77,125),(66,155,67,126)], [(1,79,63,105),(2,80,64,106),(3,81,65,107),(4,82,66,108),(5,83,56,109),(6,84,57,110),(7,85,58,100),(8,86,59,101),(9,87,60,102),(10,88,61,103),(11,78,62,104),(12,142,37,160),(13,143,38,161),(14,133,39,162),(15,134,40,163),(16,135,41,164),(17,136,42,165),(18,137,43,155),(19,138,44,156),(20,139,34,157),(21,140,35,158),(22,141,36,159),(23,153,171,127),(24,154,172,128),(25,144,173,129),(26,145,174,130),(27,146,175,131),(28,147,176,132),(29,148,166,122),(30,149,167,123),(31,150,168,124),(32,151,169,125),(33,152,170,126),(45,118,74,89),(46,119,75,90),(47,120,76,91),(48,121,77,92),(49,111,67,93),(50,112,68,94),(51,113,69,95),(52,114,70,96),(53,115,71,97),(54,116,72,98),(55,117,73,99)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,45),(12,175),(13,176),(14,166),(15,167),(16,168),(17,169),(18,170),(19,171),(20,172),(21,173),(22,174),(23,44),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77),(66,67),(78,118),(79,119),(80,120),(81,121),(82,111),(83,112),(84,113),(85,114),(86,115),(87,116),(88,117),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109),(95,110),(96,100),(97,101),(98,102),(99,103),(122,133),(123,134),(124,135),(125,136),(126,137),(127,138),(128,139),(129,140),(130,141),(131,142),(132,143),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,155),(153,156),(154,157)]])
220 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22AX | 44A | ··· | 44AN | 44AO | ··· | 44EJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
220 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C11 | C22 | C22 | C22 | C22 | C44 | C4○D4 | C11×C4○D4 |
| kernel | C11×C42⋊C2 | C4×C44 | C11×C22⋊C4 | C11×C4⋊C4 | C22×C44 | C2×C44 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 10 | 20 | 20 | 20 | 10 | 80 | 4 | 40 |
Matrix representation of C11×C42⋊C2 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 34 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 88 | 0 |
| 88 | 0 | 0 |
| 0 | 55 | 0 |
| 0 | 0 | 55 |
| 1 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(89))| [1,0,0,0,4,0,0,0,4],[34,0,0,0,0,88,0,1,0],[88,0,0,0,55,0,0,0,55],[1,0,0,0,88,0,0,0,1] >;
C11×C42⋊C2 in GAP, Magma, Sage, TeX
C_{11}\times C_4^2\rtimes C_2 % in TeX
G:=Group("C11xC4^2:C2"); // GroupNames label
G:=SmallGroup(352,152);
// by ID
G=gap.SmallGroup(352,152);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1056,1081,410]);
// 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,c*d=d*c>;
// generators/relations