direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×D4⋊C4, D4⋊1C44, C44.60D4, C22.13D8, C22.9SD16, C4⋊C4⋊1C22, (C2×C88)⋊4C2, (C2×C8)⋊2C22, (D4×C11)⋊4C4, C4.1(C2×C44), C2.1(C11×D8), C44.28(C2×C4), (D4×C22).9C2, (C2×D4).3C22, (C2×C22).46D4, C4.11(D4×C11), C2.1(C11×SD16), C22.8(D4×C11), C22.24(C22⋊C4), (C2×C44).114C22, (C11×C4⋊C4)⋊10C2, (C2×C4).17(C2×C22), C2.6(C11×C22⋊C4), SmallGroup(352,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×D4⋊C4
G = < a,b,c,d | a11=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Smallest permutation representation of C11×D4⋊C4
►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 98 133 158)(2 99 134 159)(3 89 135 160)(4 90 136 161)(5 91 137 162)(6 92 138 163)(7 93 139 164)(8 94 140 165)(9 95 141 155)(10 96 142 156)(11 97 143 157)(12 38 59 74)(13 39 60 75)(14 40 61 76)(15 41 62 77)(16 42 63 67)(17 43 64 68)(18 44 65 69)(19 34 66 70)(20 35 56 71)(21 36 57 72)(22 37 58 73)(23 84 170 48)(24 85 171 49)(25 86 172 50)(26 87 173 51)(27 88 174 52)(28 78 175 53)(29 79 176 54)(30 80 166 55)(31 81 167 45)(32 82 168 46)(33 83 169 47)(100 146 125 121)(101 147 126 111)(102 148 127 112)(103 149 128 113)(104 150 129 114)(105 151 130 115)(106 152 131 116)(107 153 132 117)(108 154 122 118)(109 144 123 119)(110 145 124 120)
(1 118)(2 119)(3 120)(4 121)(5 111)(6 112)(7 113)(8 114)(9 115)(10 116)(11 117)(12 84)(13 85)(14 86)(15 87)(16 88)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 34)(31 35)(32 36)(33 37)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 174)(68 175)(69 176)(70 166)(71 167)(72 168)(73 169)(74 170)(75 171)(76 172)(77 173)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 122)(99 123)(100 161)(101 162)(102 163)(103 164)(104 165)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(133 154)(134 144)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)(141 151)(142 152)(143 153)
(1 166 108 34)(2 167 109 35)(3 168 110 36)(4 169 100 37)(5 170 101 38)(6 171 102 39)(7 172 103 40)(8 173 104 41)(9 174 105 42)(10 175 106 43)(11 176 107 44)(12 91 84 147)(13 92 85 148)(14 93 86 149)(15 94 87 150)(16 95 88 151)(17 96 78 152)(18 97 79 153)(19 98 80 154)(20 99 81 144)(21 89 82 145)(22 90 83 146)(23 126 74 137)(24 127 75 138)(25 128 76 139)(26 129 77 140)(27 130 67 141)(28 131 68 142)(29 132 69 143)(30 122 70 133)(31 123 71 134)(32 124 72 135)(33 125 73 136)(45 119 56 159)(46 120 57 160)(47 121 58 161)(48 111 59 162)(49 112 60 163)(50 113 61 164)(51 114 62 165)(52 115 63 155)(53 116 64 156)(54 117 65 157)(55 118 66 158)
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), (1,98,133,158)(2,99,134,159)(3,89,135,160)(4,90,136,161)(5,91,137,162)(6,92,138,163)(7,93,139,164)(8,94,140,165)(9,95,141,155)(10,96,142,156)(11,97,143,157)(12,38,59,74)(13,39,60,75)(14,40,61,76)(15,41,62,77)(16,42,63,67)(17,43,64,68)(18,44,65,69)(19,34,66,70)(20,35,56,71)(21,36,57,72)(22,37,58,73)(23,84,170,48)(24,85,171,49)(25,86,172,50)(26,87,173,51)(27,88,174,52)(28,78,175,53)(29,79,176,54)(30,80,166,55)(31,81,167,45)(32,82,168,46)(33,83,169,47)(100,146,125,121)(101,147,126,111)(102,148,127,112)(103,149,128,113)(104,150,129,114)(105,151,130,115)(106,152,131,116)(107,153,132,117)(108,154,122,118)(109,144,123,119)(110,145,124,120), (1,118)(2,119)(3,120)(4,121)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,84)(13,85)(14,86)(15,87)(16,88)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,34)(31,35)(32,36)(33,37)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,174)(68,175)(69,176)(70,166)(71,167)(72,168)(73,169)(74,170)(75,171)(76,172)(77,173)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,122)(99,123)(100,161)(101,162)(102,163)(103,164)(104,165)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(133,154)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153), (1,166,108,34)(2,167,109,35)(3,168,110,36)(4,169,100,37)(5,170,101,38)(6,171,102,39)(7,172,103,40)(8,173,104,41)(9,174,105,42)(10,175,106,43)(11,176,107,44)(12,91,84,147)(13,92,85,148)(14,93,86,149)(15,94,87,150)(16,95,88,151)(17,96,78,152)(18,97,79,153)(19,98,80,154)(20,99,81,144)(21,89,82,145)(22,90,83,146)(23,126,74,137)(24,127,75,138)(25,128,76,139)(26,129,77,140)(27,130,67,141)(28,131,68,142)(29,132,69,143)(30,122,70,133)(31,123,71,134)(32,124,72,135)(33,125,73,136)(45,119,56,159)(46,120,57,160)(47,121,58,161)(48,111,59,162)(49,112,60,163)(50,113,61,164)(51,114,62,165)(52,115,63,155)(53,116,64,156)(54,117,65,157)(55,118,66,158)>;
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), (1,98,133,158)(2,99,134,159)(3,89,135,160)(4,90,136,161)(5,91,137,162)(6,92,138,163)(7,93,139,164)(8,94,140,165)(9,95,141,155)(10,96,142,156)(11,97,143,157)(12,38,59,74)(13,39,60,75)(14,40,61,76)(15,41,62,77)(16,42,63,67)(17,43,64,68)(18,44,65,69)(19,34,66,70)(20,35,56,71)(21,36,57,72)(22,37,58,73)(23,84,170,48)(24,85,171,49)(25,86,172,50)(26,87,173,51)(27,88,174,52)(28,78,175,53)(29,79,176,54)(30,80,166,55)(31,81,167,45)(32,82,168,46)(33,83,169,47)(100,146,125,121)(101,147,126,111)(102,148,127,112)(103,149,128,113)(104,150,129,114)(105,151,130,115)(106,152,131,116)(107,153,132,117)(108,154,122,118)(109,144,123,119)(110,145,124,120), (1,118)(2,119)(3,120)(4,121)(5,111)(6,112)(7,113)(8,114)(9,115)(10,116)(11,117)(12,84)(13,85)(14,86)(15,87)(16,88)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,34)(31,35)(32,36)(33,37)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,174)(68,175)(69,176)(70,166)(71,167)(72,168)(73,169)(74,170)(75,171)(76,172)(77,173)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,122)(99,123)(100,161)(101,162)(102,163)(103,164)(104,165)(105,155)(106,156)(107,157)(108,158)(109,159)(110,160)(133,154)(134,144)(135,145)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153), (1,166,108,34)(2,167,109,35)(3,168,110,36)(4,169,100,37)(5,170,101,38)(6,171,102,39)(7,172,103,40)(8,173,104,41)(9,174,105,42)(10,175,106,43)(11,176,107,44)(12,91,84,147)(13,92,85,148)(14,93,86,149)(15,94,87,150)(16,95,88,151)(17,96,78,152)(18,97,79,153)(19,98,80,154)(20,99,81,144)(21,89,82,145)(22,90,83,146)(23,126,74,137)(24,127,75,138)(25,128,76,139)(26,129,77,140)(27,130,67,141)(28,131,68,142)(29,132,69,143)(30,122,70,133)(31,123,71,134)(32,124,72,135)(33,125,73,136)(45,119,56,159)(46,120,57,160)(47,121,58,161)(48,111,59,162)(49,112,60,163)(50,113,61,164)(51,114,62,165)(52,115,63,155)(53,116,64,156)(54,117,65,157)(55,118,66,158) );
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)], [(1,98,133,158),(2,99,134,159),(3,89,135,160),(4,90,136,161),(5,91,137,162),(6,92,138,163),(7,93,139,164),(8,94,140,165),(9,95,141,155),(10,96,142,156),(11,97,143,157),(12,38,59,74),(13,39,60,75),(14,40,61,76),(15,41,62,77),(16,42,63,67),(17,43,64,68),(18,44,65,69),(19,34,66,70),(20,35,56,71),(21,36,57,72),(22,37,58,73),(23,84,170,48),(24,85,171,49),(25,86,172,50),(26,87,173,51),(27,88,174,52),(28,78,175,53),(29,79,176,54),(30,80,166,55),(31,81,167,45),(32,82,168,46),(33,83,169,47),(100,146,125,121),(101,147,126,111),(102,148,127,112),(103,149,128,113),(104,150,129,114),(105,151,130,115),(106,152,131,116),(107,153,132,117),(108,154,122,118),(109,144,123,119),(110,145,124,120)], [(1,118),(2,119),(3,120),(4,121),(5,111),(6,112),(7,113),(8,114),(9,115),(10,116),(11,117),(12,84),(13,85),(14,86),(15,87),(16,88),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(29,44),(30,34),(31,35),(32,36),(33,37),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(67,174),(68,175),(69,176),(70,166),(71,167),(72,168),(73,169),(74,170),(75,171),(76,172),(77,173),(89,124),(90,125),(91,126),(92,127),(93,128),(94,129),(95,130),(96,131),(97,132),(98,122),(99,123),(100,161),(101,162),(102,163),(103,164),(104,165),(105,155),(106,156),(107,157),(108,158),(109,159),(110,160),(133,154),(134,144),(135,145),(136,146),(137,147),(138,148),(139,149),(140,150),(141,151),(142,152),(143,153)], [(1,166,108,34),(2,167,109,35),(3,168,110,36),(4,169,100,37),(5,170,101,38),(6,171,102,39),(7,172,103,40),(8,173,104,41),(9,174,105,42),(10,175,106,43),(11,176,107,44),(12,91,84,147),(13,92,85,148),(14,93,86,149),(15,94,87,150),(16,95,88,151),(17,96,78,152),(18,97,79,153),(19,98,80,154),(20,99,81,144),(21,89,82,145),(22,90,83,146),(23,126,74,137),(24,127,75,138),(25,128,76,139),(26,129,77,140),(27,130,67,141),(28,131,68,142),(29,132,69,143),(30,122,70,133),(31,123,71,134),(32,124,72,135),(33,125,73,136),(45,119,56,159),(46,120,57,160),(47,121,58,161),(48,111,59,162),(49,112,60,163),(50,113,61,164),(51,114,62,165),(52,115,63,155),(53,116,64,156),(54,117,65,157),(55,118,66,158)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22AX | 44A | ··· | 44T | 44U | ··· | 44AN | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C11 | C22 | C22 | C22 | C44 | D4 | D4 | D8 | SD16 | D4×C11 | D4×C11 | C11×D8 | C11×SD16 |
| kernel | C11×D4⋊C4 | C11×C4⋊C4 | C2×C88 | D4×C22 | D4×C11 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C44 | C2×C22 | C22 | C22 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 40 | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 |
Matrix representation of C11×D4⋊C4 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 1 | 87 |
| 0 | 1 | 88 |
| 1 | 0 | 0 |
| 0 | 1 | 87 |
| 0 | 0 | 88 |
| 34 | 0 | 0 |
| 0 | 0 | 25 |
| 0 | 57 | 0 |
G:=sub<GL(3,GF(89))| [1,0,0,0,8,0,0,0,8],[1,0,0,0,1,1,0,87,88],[1,0,0,0,1,0,0,87,88],[34,0,0,0,0,57,0,25,0] >;
C11×D4⋊C4 in GAP, Magma, Sage, TeX
C_{11}\times D_4\rtimes C_4 % in TeX
G:=Group("C11xD4:C4"); // GroupNames label
G:=SmallGroup(352,51);
// by ID
G=gap.SmallGroup(352,51);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,5283,2649,117]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations
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