direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C4⋊D4, C44⋊9D4, C4⋊C4⋊2C22, (C2×C22)⋊4D4, C4⋊2(D4×C11), (C2×D4)⋊2C22, C2.5(D4×C22), (D4×C22)⋊11C2, C22⋊C4⋊3C22, (C22×C4)⋊4C22, C22.68(C2×D4), C22⋊1(D4×C11), (C22×C44)⋊11C2, C23.8(C2×C22), C22.41(C4○D4), (C2×C22).76C23, (C2×C44).123C22, (C22×C22).27C22, C22.11(C22×C22), (C11×C4⋊C4)⋊11C2, (C2×C4).3(C2×C22), C2.4(C11×C4○D4), (C11×C22⋊C4)⋊11C2, SmallGroup(352,156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C4⋊D4
G = < a,b,c,d | a11=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C22, C22, C4⋊D4, C44, C44, C2×C22, C2×C22, C2×C22, C2×C44, C2×C44, C2×C44, D4×C11, C22×C22, C22×C22, C11×C22⋊C4, C11×C4⋊C4, C22×C44, D4×C22, D4×C22, C11×C4⋊D4
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C4○D4, C22, C4⋊D4, C2×C22, D4×C11, C22×C22, D4×C22, C11×C4○D4, C11×C4⋊D4
►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)
(1 123 49 137)(2 124 50 138)(3 125 51 139)(4 126 52 140)(5 127 53 141)(6 128 54 142)(7 129 55 143)(8 130 45 133)(9 131 46 134)(10 132 47 135)(11 122 48 136)(12 91 176 88)(13 92 166 78)(14 93 167 79)(15 94 168 80)(16 95 169 81)(17 96 170 82)(18 97 171 83)(19 98 172 84)(20 99 173 85)(21 89 174 86)(22 90 175 87)(23 121 36 110)(24 111 37 100)(25 112 38 101)(26 113 39 102)(27 114 40 103)(28 115 41 104)(29 116 42 105)(30 117 43 106)(31 118 44 107)(32 119 34 108)(33 120 35 109)(56 155 67 144)(57 156 68 145)(58 157 69 146)(59 158 70 147)(60 159 71 148)(61 160 72 149)(62 161 73 150)(63 162 74 151)(64 163 75 152)(65 164 76 153)(66 165 77 154)
(1 93 70 114)(2 94 71 115)(3 95 72 116)(4 96 73 117)(5 97 74 118)(6 98 75 119)(7 99 76 120)(8 89 77 121)(9 90 67 111)(10 91 68 112)(11 92 69 113)(12 145 25 132)(13 146 26 122)(14 147 27 123)(15 148 28 124)(16 149 29 125)(17 150 30 126)(18 151 31 127)(19 152 32 128)(20 153 33 129)(21 154 23 130)(22 144 24 131)(34 142 172 163)(35 143 173 164)(36 133 174 165)(37 134 175 155)(38 135 176 156)(39 136 166 157)(40 137 167 158)(41 138 168 159)(42 139 169 160)(43 140 170 161)(44 141 171 162)(45 86 66 110)(46 87 56 100)(47 88 57 101)(48 78 58 102)(49 79 59 103)(50 80 60 104)(51 81 61 105)(52 82 62 106)(53 83 63 107)(54 84 64 108)(55 85 65 109)
(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)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 100)(88 101)(89 121)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(122 136)(123 137)(124 138)(125 139)(126 140)(127 141)(128 142)(129 143)(130 133)(131 134)(132 135)(144 155)(145 156)(146 157)(147 158)(148 159)(149 160)(150 161)(151 162)(152 163)(153 164)(154 165)
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), (1,123,49,137)(2,124,50,138)(3,125,51,139)(4,126,52,140)(5,127,53,141)(6,128,54,142)(7,129,55,143)(8,130,45,133)(9,131,46,134)(10,132,47,135)(11,122,48,136)(12,91,176,88)(13,92,166,78)(14,93,167,79)(15,94,168,80)(16,95,169,81)(17,96,170,82)(18,97,171,83)(19,98,172,84)(20,99,173,85)(21,89,174,86)(22,90,175,87)(23,121,36,110)(24,111,37,100)(25,112,38,101)(26,113,39,102)(27,114,40,103)(28,115,41,104)(29,116,42,105)(30,117,43,106)(31,118,44,107)(32,119,34,108)(33,120,35,109)(56,155,67,144)(57,156,68,145)(58,157,69,146)(59,158,70,147)(60,159,71,148)(61,160,72,149)(62,161,73,150)(63,162,74,151)(64,163,75,152)(65,164,76,153)(66,165,77,154), (1,93,70,114)(2,94,71,115)(3,95,72,116)(4,96,73,117)(5,97,74,118)(6,98,75,119)(7,99,76,120)(8,89,77,121)(9,90,67,111)(10,91,68,112)(11,92,69,113)(12,145,25,132)(13,146,26,122)(14,147,27,123)(15,148,28,124)(16,149,29,125)(17,150,30,126)(18,151,31,127)(19,152,32,128)(20,153,33,129)(21,154,23,130)(22,144,24,131)(34,142,172,163)(35,143,173,164)(36,133,174,165)(37,134,175,155)(38,135,176,156)(39,136,166,157)(40,137,167,158)(41,138,168,159)(42,139,169,160)(43,140,170,161)(44,141,171,162)(45,86,66,110)(46,87,56,100)(47,88,57,101)(48,78,58,102)(49,79,59,103)(50,80,60,104)(51,81,61,105)(52,82,62,106)(53,83,63,107)(54,84,64,108)(55,85,65,109), (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)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,100)(88,101)(89,121)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(129,143)(130,133)(131,134)(132,135)(144,155)(145,156)(146,157)(147,158)(148,159)(149,160)(150,161)(151,162)(152,163)(153,164)(154,165)>;
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), (1,123,49,137)(2,124,50,138)(3,125,51,139)(4,126,52,140)(5,127,53,141)(6,128,54,142)(7,129,55,143)(8,130,45,133)(9,131,46,134)(10,132,47,135)(11,122,48,136)(12,91,176,88)(13,92,166,78)(14,93,167,79)(15,94,168,80)(16,95,169,81)(17,96,170,82)(18,97,171,83)(19,98,172,84)(20,99,173,85)(21,89,174,86)(22,90,175,87)(23,121,36,110)(24,111,37,100)(25,112,38,101)(26,113,39,102)(27,114,40,103)(28,115,41,104)(29,116,42,105)(30,117,43,106)(31,118,44,107)(32,119,34,108)(33,120,35,109)(56,155,67,144)(57,156,68,145)(58,157,69,146)(59,158,70,147)(60,159,71,148)(61,160,72,149)(62,161,73,150)(63,162,74,151)(64,163,75,152)(65,164,76,153)(66,165,77,154), (1,93,70,114)(2,94,71,115)(3,95,72,116)(4,96,73,117)(5,97,74,118)(6,98,75,119)(7,99,76,120)(8,89,77,121)(9,90,67,111)(10,91,68,112)(11,92,69,113)(12,145,25,132)(13,146,26,122)(14,147,27,123)(15,148,28,124)(16,149,29,125)(17,150,30,126)(18,151,31,127)(19,152,32,128)(20,153,33,129)(21,154,23,130)(22,144,24,131)(34,142,172,163)(35,143,173,164)(36,133,174,165)(37,134,175,155)(38,135,176,156)(39,136,166,157)(40,137,167,158)(41,138,168,159)(42,139,169,160)(43,140,170,161)(44,141,171,162)(45,86,66,110)(46,87,56,100)(47,88,57,101)(48,78,58,102)(49,79,59,103)(50,80,60,104)(51,81,61,105)(52,82,62,106)(53,83,63,107)(54,84,64,108)(55,85,65,109), (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)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,100)(88,101)(89,121)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(129,143)(130,133)(131,134)(132,135)(144,155)(145,156)(146,157)(147,158)(148,159)(149,160)(150,161)(151,162)(152,163)(153,164)(154,165) );
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)], [(1,123,49,137),(2,124,50,138),(3,125,51,139),(4,126,52,140),(5,127,53,141),(6,128,54,142),(7,129,55,143),(8,130,45,133),(9,131,46,134),(10,132,47,135),(11,122,48,136),(12,91,176,88),(13,92,166,78),(14,93,167,79),(15,94,168,80),(16,95,169,81),(17,96,170,82),(18,97,171,83),(19,98,172,84),(20,99,173,85),(21,89,174,86),(22,90,175,87),(23,121,36,110),(24,111,37,100),(25,112,38,101),(26,113,39,102),(27,114,40,103),(28,115,41,104),(29,116,42,105),(30,117,43,106),(31,118,44,107),(32,119,34,108),(33,120,35,109),(56,155,67,144),(57,156,68,145),(58,157,69,146),(59,158,70,147),(60,159,71,148),(61,160,72,149),(62,161,73,150),(63,162,74,151),(64,163,75,152),(65,164,76,153),(66,165,77,154)], [(1,93,70,114),(2,94,71,115),(3,95,72,116),(4,96,73,117),(5,97,74,118),(6,98,75,119),(7,99,76,120),(8,89,77,121),(9,90,67,111),(10,91,68,112),(11,92,69,113),(12,145,25,132),(13,146,26,122),(14,147,27,123),(15,148,28,124),(16,149,29,125),(17,150,30,126),(18,151,31,127),(19,152,32,128),(20,153,33,129),(21,154,23,130),(22,144,24,131),(34,142,172,163),(35,143,173,164),(36,133,174,165),(37,134,175,155),(38,135,176,156),(39,136,166,157),(40,137,167,158),(41,138,168,159),(42,139,169,160),(43,140,170,161),(44,141,171,162),(45,86,66,110),(46,87,56,100),(47,88,57,101),(48,78,58,102),(49,79,59,103),(50,80,60,104),(51,81,61,105),(52,82,62,106),(53,83,63,107),(54,84,64,108),(55,85,65,109)], [(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),(78,102),(79,103),(80,104),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,100),(88,101),(89,121),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,118),(98,119),(99,120),(122,136),(123,137),(124,138),(125,139),(126,140),(127,141),(128,142),(129,143),(130,133),(131,134),(132,135),(144,155),(145,156),(146,157),(147,158),(148,159),(149,160),(150,161),(151,162),(152,163),(153,164),(154,165)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22AX | 22AY | ··· | 22BR | 44A | ··· | 44AN | 44AO | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | C22 | D4 | D4 | C4○D4 | D4×C11 | D4×C11 | C11×C4○D4 |
| kernel | C11×C4⋊D4 | C11×C22⋊C4 | C11×C4⋊C4 | C22×C44 | D4×C22 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C44 | C2×C22 | C22 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 10 | 20 | 10 | 10 | 30 | 2 | 2 | 2 | 20 | 20 | 20 |
Matrix representation of C11×C4⋊D4 ►in GL4(𝔽89) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 34 | 85 | 0 | 0 |
| 0 | 55 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 88 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 17 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 88 |
G:=sub<GL(4,GF(89))| [2,0,0,0,0,2,0,0,0,0,8,0,0,0,0,8],[34,0,0,0,85,55,0,0,0,0,88,0,0,0,0,88],[88,72,0,0,0,1,0,0,0,0,0,1,0,0,88,0],[1,17,0,0,0,88,0,0,0,0,1,0,0,0,0,88] >;
C11×C4⋊D4 in GAP, Magma, Sage, TeX
C_{11}\times C_4\rtimes D_4 % in TeX
G:=Group("C11xC4:D4"); // GroupNames label
G:=SmallGroup(352,156);
// by ID
G=gap.SmallGroup(352,156);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,535,3242]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations