direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C11×C4⋊1D4, C44⋊6D4, C42⋊6C22, C4⋊1(D4×C11), (C4×C44)⋊13C2, (C2×D4)⋊3C22, C2.9(D4×C22), (D4×C22)⋊12C2, C22.72(C2×D4), C23.4(C2×C22), (C2×C22).82C23, (C2×C44).125C22, (C22×C22).4C22, C22.17(C22×C22), (C2×C4).23(C2×C22), SmallGroup(352,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C4⋊1D4
G = < a,b,c,d | a11=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C11, C42, C2×D4, C22, C22, C4⋊1D4, C44, C2×C22, C2×C22, C2×C44, D4×C11, C22×C22, C4×C44, D4×C22, C11×C4⋊1D4
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C22, C4⋊1D4, C2×C22, D4×C11, C22×C22, D4×C22, C11×C4⋊1D4
►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)
(1 167 66 27)(2 168 56 28)(3 169 57 29)(4 170 58 30)(5 171 59 31)(6 172 60 32)(7 173 61 33)(8 174 62 23)(9 175 63 24)(10 176 64 25)(11 166 65 26)(12 72 42 52)(13 73 43 53)(14 74 44 54)(15 75 34 55)(16 76 35 45)(17 77 36 46)(18 67 37 47)(19 68 38 48)(20 69 39 49)(21 70 40 50)(22 71 41 51)(78 157 113 122)(79 158 114 123)(80 159 115 124)(81 160 116 125)(82 161 117 126)(83 162 118 127)(84 163 119 128)(85 164 120 129)(86 165 121 130)(87 155 111 131)(88 156 112 132)(89 150 106 133)(90 151 107 134)(91 152 108 135)(92 153 109 136)(93 154 110 137)(94 144 100 138)(95 145 101 139)(96 146 102 140)(97 147 103 141)(98 148 104 142)(99 149 105 143)
(1 114 70 93)(2 115 71 94)(3 116 72 95)(4 117 73 96)(5 118 74 97)(6 119 75 98)(7 120 76 99)(8 121 77 89)(9 111 67 90)(10 112 68 91)(11 113 69 92)(12 139 29 160)(13 140 30 161)(14 141 31 162)(15 142 32 163)(16 143 33 164)(17 133 23 165)(18 134 24 155)(19 135 25 156)(20 136 26 157)(21 137 27 158)(22 138 28 159)(34 148 172 128)(35 149 173 129)(36 150 174 130)(37 151 175 131)(38 152 176 132)(39 153 166 122)(40 154 167 123)(41 144 168 124)(42 145 169 125)(43 146 170 126)(44 147 171 127)(45 105 61 85)(46 106 62 86)(47 107 63 87)(48 108 64 88)(49 109 65 78)(50 110 66 79)(51 100 56 80)(52 101 57 81)(53 102 58 82)(54 103 59 83)(55 104 60 84)
(1 158)(2 159)(3 160)(4 161)(5 162)(6 163)(7 164)(8 165)(9 155)(10 156)(11 157)(12 95)(13 96)(14 97)(15 98)(16 99)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 121)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 100)(42 101)(43 102)(44 103)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 144)(52 145)(53 146)(54 147)(55 148)(56 124)(57 125)(58 126)(59 127)(60 128)(61 129)(62 130)(63 131)(64 132)(65 122)(66 123)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 141)(75 142)(76 143)(77 133)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)
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), (1,167,66,27)(2,168,56,28)(3,169,57,29)(4,170,58,30)(5,171,59,31)(6,172,60,32)(7,173,61,33)(8,174,62,23)(9,175,63,24)(10,176,64,25)(11,166,65,26)(12,72,42,52)(13,73,43,53)(14,74,44,54)(15,75,34,55)(16,76,35,45)(17,77,36,46)(18,67,37,47)(19,68,38,48)(20,69,39,49)(21,70,40,50)(22,71,41,51)(78,157,113,122)(79,158,114,123)(80,159,115,124)(81,160,116,125)(82,161,117,126)(83,162,118,127)(84,163,119,128)(85,164,120,129)(86,165,121,130)(87,155,111,131)(88,156,112,132)(89,150,106,133)(90,151,107,134)(91,152,108,135)(92,153,109,136)(93,154,110,137)(94,144,100,138)(95,145,101,139)(96,146,102,140)(97,147,103,141)(98,148,104,142)(99,149,105,143), (1,114,70,93)(2,115,71,94)(3,116,72,95)(4,117,73,96)(5,118,74,97)(6,119,75,98)(7,120,76,99)(8,121,77,89)(9,111,67,90)(10,112,68,91)(11,113,69,92)(12,139,29,160)(13,140,30,161)(14,141,31,162)(15,142,32,163)(16,143,33,164)(17,133,23,165)(18,134,24,155)(19,135,25,156)(20,136,26,157)(21,137,27,158)(22,138,28,159)(34,148,172,128)(35,149,173,129)(36,150,174,130)(37,151,175,131)(38,152,176,132)(39,153,166,122)(40,154,167,123)(41,144,168,124)(42,145,169,125)(43,146,170,126)(44,147,171,127)(45,105,61,85)(46,106,62,86)(47,107,63,87)(48,108,64,88)(49,109,65,78)(50,110,66,79)(51,100,56,80)(52,101,57,81)(53,102,58,82)(54,103,59,83)(55,104,60,84), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,155)(10,156)(11,157)(12,95)(13,96)(14,97)(15,98)(16,99)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,121)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,100)(42,101)(43,102)(44,103)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,144)(52,145)(53,146)(54,147)(55,148)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,122)(66,123)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,133)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176)>;
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), (1,167,66,27)(2,168,56,28)(3,169,57,29)(4,170,58,30)(5,171,59,31)(6,172,60,32)(7,173,61,33)(8,174,62,23)(9,175,63,24)(10,176,64,25)(11,166,65,26)(12,72,42,52)(13,73,43,53)(14,74,44,54)(15,75,34,55)(16,76,35,45)(17,77,36,46)(18,67,37,47)(19,68,38,48)(20,69,39,49)(21,70,40,50)(22,71,41,51)(78,157,113,122)(79,158,114,123)(80,159,115,124)(81,160,116,125)(82,161,117,126)(83,162,118,127)(84,163,119,128)(85,164,120,129)(86,165,121,130)(87,155,111,131)(88,156,112,132)(89,150,106,133)(90,151,107,134)(91,152,108,135)(92,153,109,136)(93,154,110,137)(94,144,100,138)(95,145,101,139)(96,146,102,140)(97,147,103,141)(98,148,104,142)(99,149,105,143), (1,114,70,93)(2,115,71,94)(3,116,72,95)(4,117,73,96)(5,118,74,97)(6,119,75,98)(7,120,76,99)(8,121,77,89)(9,111,67,90)(10,112,68,91)(11,113,69,92)(12,139,29,160)(13,140,30,161)(14,141,31,162)(15,142,32,163)(16,143,33,164)(17,133,23,165)(18,134,24,155)(19,135,25,156)(20,136,26,157)(21,137,27,158)(22,138,28,159)(34,148,172,128)(35,149,173,129)(36,150,174,130)(37,151,175,131)(38,152,176,132)(39,153,166,122)(40,154,167,123)(41,144,168,124)(42,145,169,125)(43,146,170,126)(44,147,171,127)(45,105,61,85)(46,106,62,86)(47,107,63,87)(48,108,64,88)(49,109,65,78)(50,110,66,79)(51,100,56,80)(52,101,57,81)(53,102,58,82)(54,103,59,83)(55,104,60,84), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,155)(10,156)(11,157)(12,95)(13,96)(14,97)(15,98)(16,99)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,121)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,100)(42,101)(43,102)(44,103)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,144)(52,145)(53,146)(54,147)(55,148)(56,124)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,122)(66,123)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,133)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176) );
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)], [(1,167,66,27),(2,168,56,28),(3,169,57,29),(4,170,58,30),(5,171,59,31),(6,172,60,32),(7,173,61,33),(8,174,62,23),(9,175,63,24),(10,176,64,25),(11,166,65,26),(12,72,42,52),(13,73,43,53),(14,74,44,54),(15,75,34,55),(16,76,35,45),(17,77,36,46),(18,67,37,47),(19,68,38,48),(20,69,39,49),(21,70,40,50),(22,71,41,51),(78,157,113,122),(79,158,114,123),(80,159,115,124),(81,160,116,125),(82,161,117,126),(83,162,118,127),(84,163,119,128),(85,164,120,129),(86,165,121,130),(87,155,111,131),(88,156,112,132),(89,150,106,133),(90,151,107,134),(91,152,108,135),(92,153,109,136),(93,154,110,137),(94,144,100,138),(95,145,101,139),(96,146,102,140),(97,147,103,141),(98,148,104,142),(99,149,105,143)], [(1,114,70,93),(2,115,71,94),(3,116,72,95),(4,117,73,96),(5,118,74,97),(6,119,75,98),(7,120,76,99),(8,121,77,89),(9,111,67,90),(10,112,68,91),(11,113,69,92),(12,139,29,160),(13,140,30,161),(14,141,31,162),(15,142,32,163),(16,143,33,164),(17,133,23,165),(18,134,24,155),(19,135,25,156),(20,136,26,157),(21,137,27,158),(22,138,28,159),(34,148,172,128),(35,149,173,129),(36,150,174,130),(37,151,175,131),(38,152,176,132),(39,153,166,122),(40,154,167,123),(41,144,168,124),(42,145,169,125),(43,146,170,126),(44,147,171,127),(45,105,61,85),(46,106,62,86),(47,107,63,87),(48,108,64,88),(49,109,65,78),(50,110,66,79),(51,100,56,80),(52,101,57,81),(53,102,58,82),(54,103,59,83),(55,104,60,84)], [(1,158),(2,159),(3,160),(4,161),(5,162),(6,163),(7,164),(8,165),(9,155),(10,156),(11,157),(12,95),(13,96),(14,97),(15,98),(16,99),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,121),(24,111),(25,112),(26,113),(27,114),(28,115),(29,116),(30,117),(31,118),(32,119),(33,120),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,100),(42,101),(43,102),(44,103),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,144),(52,145),(53,146),(54,147),(55,148),(56,124),(57,125),(58,126),(59,127),(60,128),(61,129),(62,130),(63,131),(64,132),(65,122),(66,123),(67,134),(68,135),(69,136),(70,137),(71,138),(72,139),(73,140),(74,141),(75,142),(76,143),(77,133),(78,166),(79,167),(80,168),(81,169),(82,170),(83,171),(84,172),(85,173),(86,174),(87,175),(88,176)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22BR | 44A | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D4 | D4×C11 |
| kernel | C11×C4⋊1D4 | C4×C44 | D4×C22 | C4⋊1D4 | C42 | C2×D4 | C44 | C4 |
| # reps | 1 | 1 | 6 | 10 | 10 | 60 | 6 | 60 |
Matrix representation of C11×C4⋊1D4 ►in GL4(𝔽89) generated by
| 67 | 0 | 0 | 0 |
| 0 | 67 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 88 | 0 |
| 0 | 1 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 88 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(89))| [67,0,0,0,0,67,0,0,0,0,1,0,0,0,0,1],[88,0,0,0,0,88,0,0,0,0,0,88,0,0,1,0],[0,88,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,88,0,0,88,0,0,0,0,0,0,1,0,0,1,0] >;
C11×C4⋊1D4 in GAP, Magma, Sage, TeX
C_{11}\times C_4\rtimes_1D_4 % in TeX
G:=Group("C11xC4:1D4"); // GroupNames label
G:=SmallGroup(352,162);
// by ID
G=gap.SmallGroup(352,162);
# by ID
G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,535,3242,806]);
// 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^-1,d*c*d=c^-1>;
// generators/relations