metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1D11, (C4×C44)⋊9C2, (C4×D11)⋊3C4, C44.25(C2×C4), D22⋊C4.7C2, D22.3(C2×C4), C4.22(C4×D11), (C2×C4).96D22, (C4×Dic11)⋊8C2, C22.3(C4○D4), C22.3(C22×C4), Dic11⋊C4⋊17C2, C11⋊1(C42⋊C2), (C2×C22).13C23, (C2×C44).73C22, Dic11.5(C2×C4), C2.2(D44⋊5C2), C22.10(C22×D11), (C2×Dic11).24C22, (C22×D11).14C22, C2.5(C2×C4×D11), (C2×C4×D11).8C2, SmallGroup(352,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊D11
G = < a,b,c,d | a4=b4=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 402 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, C23, C11, C42, C42, C22⋊C4, C4⋊C4, C22×C4, D11, C22, C22, C42⋊C2, Dic11, Dic11, C44, C44, D22, D22, C2×C22, C4×D11, C2×Dic11, C2×Dic11, C2×C44, C2×C44, C22×D11, C4×Dic11, Dic11⋊C4, D22⋊C4, C4×C44, C2×C4×D11, C42⋊D11
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, D11, C42⋊C2, D22, C4×D11, C22×D11, C2×C4×D11, D44⋊5C2, C42⋊D11
►On 176 points
Generators in S176
(1 109 21 98)(2 110 22 99)(3 100 12 89)(4 101 13 90)(5 102 14 91)(6 103 15 92)(7 104 16 93)(8 105 17 94)(9 106 18 95)(10 107 19 96)(11 108 20 97)(23 122 34 111)(24 123 35 112)(25 124 36 113)(26 125 37 114)(27 126 38 115)(28 127 39 116)(29 128 40 117)(30 129 41 118)(31 130 42 119)(32 131 43 120)(33 132 44 121)(45 144 56 133)(46 145 57 134)(47 146 58 135)(48 147 59 136)(49 148 60 137)(50 149 61 138)(51 150 62 139)(52 151 63 140)(53 152 64 141)(54 153 65 142)(55 154 66 143)(67 166 78 155)(68 167 79 156)(69 168 80 157)(70 169 81 158)(71 170 82 159)(72 171 83 160)(73 172 84 161)(74 173 85 162)(75 174 86 163)(76 175 87 164)(77 176 88 165)
(1 76 32 54)(2 77 33 55)(3 67 23 45)(4 68 24 46)(5 69 25 47)(6 70 26 48)(7 71 27 49)(8 72 28 50)(9 73 29 51)(10 74 30 52)(11 75 31 53)(12 78 34 56)(13 79 35 57)(14 80 36 58)(15 81 37 59)(16 82 38 60)(17 83 39 61)(18 84 40 62)(19 85 41 63)(20 86 42 64)(21 87 43 65)(22 88 44 66)(89 155 111 133)(90 156 112 134)(91 157 113 135)(92 158 114 136)(93 159 115 137)(94 160 116 138)(95 161 117 139)(96 162 118 140)(97 163 119 141)(98 164 120 142)(99 165 121 143)(100 166 122 144)(101 167 123 145)(102 168 124 146)(103 169 125 147)(104 170 126 148)(105 171 127 149)(106 172 128 150)(107 173 129 151)(108 174 130 152)(109 175 131 153)(110 176 132 154)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)(89 95)(90 94)(91 93)(96 99)(97 98)(100 106)(101 105)(102 104)(107 110)(108 109)(111 117)(112 116)(113 115)(118 121)(119 120)(122 128)(123 127)(124 126)(129 132)(130 131)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 154)(141 153)(142 152)(143 151)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 176)(163 175)(164 174)(165 173)
G:=sub<Sym(176)| (1,109,21,98)(2,110,22,99)(3,100,12,89)(4,101,13,90)(5,102,14,91)(6,103,15,92)(7,104,16,93)(8,105,17,94)(9,106,18,95)(10,107,19,96)(11,108,20,97)(23,122,34,111)(24,123,35,112)(25,124,36,113)(26,125,37,114)(27,126,38,115)(28,127,39,116)(29,128,40,117)(30,129,41,118)(31,130,42,119)(32,131,43,120)(33,132,44,121)(45,144,56,133)(46,145,57,134)(47,146,58,135)(48,147,59,136)(49,148,60,137)(50,149,61,138)(51,150,62,139)(52,151,63,140)(53,152,64,141)(54,153,65,142)(55,154,66,143)(67,166,78,155)(68,167,79,156)(69,168,80,157)(70,169,81,158)(71,170,82,159)(72,171,83,160)(73,172,84,161)(74,173,85,162)(75,174,86,163)(76,175,87,164)(77,176,88,165), (1,76,32,54)(2,77,33,55)(3,67,23,45)(4,68,24,46)(5,69,25,47)(6,70,26,48)(7,71,27,49)(8,72,28,50)(9,73,29,51)(10,74,30,52)(11,75,31,53)(12,78,34,56)(13,79,35,57)(14,80,36,58)(15,81,37,59)(16,82,38,60)(17,83,39,61)(18,84,40,62)(19,85,41,63)(20,86,42,64)(21,87,43,65)(22,88,44,66)(89,155,111,133)(90,156,112,134)(91,157,113,135)(92,158,114,136)(93,159,115,137)(94,160,116,138)(95,161,117,139)(96,162,118,140)(97,163,119,141)(98,164,120,142)(99,165,121,143)(100,166,122,144)(101,167,123,145)(102,168,124,146)(103,169,125,147)(104,170,126,148)(105,171,127,149)(106,172,128,150)(107,173,129,151)(108,174,130,152)(109,175,131,153)(110,176,132,154), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)(89,95)(90,94)(91,93)(96,99)(97,98)(100,106)(101,105)(102,104)(107,110)(108,109)(111,117)(112,116)(113,115)(118,121)(119,120)(122,128)(123,127)(124,126)(129,132)(130,131)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,154)(141,153)(142,152)(143,151)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,176)(163,175)(164,174)(165,173)>;
G:=Group( (1,109,21,98)(2,110,22,99)(3,100,12,89)(4,101,13,90)(5,102,14,91)(6,103,15,92)(7,104,16,93)(8,105,17,94)(9,106,18,95)(10,107,19,96)(11,108,20,97)(23,122,34,111)(24,123,35,112)(25,124,36,113)(26,125,37,114)(27,126,38,115)(28,127,39,116)(29,128,40,117)(30,129,41,118)(31,130,42,119)(32,131,43,120)(33,132,44,121)(45,144,56,133)(46,145,57,134)(47,146,58,135)(48,147,59,136)(49,148,60,137)(50,149,61,138)(51,150,62,139)(52,151,63,140)(53,152,64,141)(54,153,65,142)(55,154,66,143)(67,166,78,155)(68,167,79,156)(69,168,80,157)(70,169,81,158)(71,170,82,159)(72,171,83,160)(73,172,84,161)(74,173,85,162)(75,174,86,163)(76,175,87,164)(77,176,88,165), (1,76,32,54)(2,77,33,55)(3,67,23,45)(4,68,24,46)(5,69,25,47)(6,70,26,48)(7,71,27,49)(8,72,28,50)(9,73,29,51)(10,74,30,52)(11,75,31,53)(12,78,34,56)(13,79,35,57)(14,80,36,58)(15,81,37,59)(16,82,38,60)(17,83,39,61)(18,84,40,62)(19,85,41,63)(20,86,42,64)(21,87,43,65)(22,88,44,66)(89,155,111,133)(90,156,112,134)(91,157,113,135)(92,158,114,136)(93,159,115,137)(94,160,116,138)(95,161,117,139)(96,162,118,140)(97,163,119,141)(98,164,120,142)(99,165,121,143)(100,166,122,144)(101,167,123,145)(102,168,124,146)(103,169,125,147)(104,170,126,148)(105,171,127,149)(106,172,128,150)(107,173,129,151)(108,174,130,152)(109,175,131,153)(110,176,132,154), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)(89,95)(90,94)(91,93)(96,99)(97,98)(100,106)(101,105)(102,104)(107,110)(108,109)(111,117)(112,116)(113,115)(118,121)(119,120)(122,128)(123,127)(124,126)(129,132)(130,131)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,154)(141,153)(142,152)(143,151)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,176)(163,175)(164,174)(165,173) );
G=PermutationGroup([[(1,109,21,98),(2,110,22,99),(3,100,12,89),(4,101,13,90),(5,102,14,91),(6,103,15,92),(7,104,16,93),(8,105,17,94),(9,106,18,95),(10,107,19,96),(11,108,20,97),(23,122,34,111),(24,123,35,112),(25,124,36,113),(26,125,37,114),(27,126,38,115),(28,127,39,116),(29,128,40,117),(30,129,41,118),(31,130,42,119),(32,131,43,120),(33,132,44,121),(45,144,56,133),(46,145,57,134),(47,146,58,135),(48,147,59,136),(49,148,60,137),(50,149,61,138),(51,150,62,139),(52,151,63,140),(53,152,64,141),(54,153,65,142),(55,154,66,143),(67,166,78,155),(68,167,79,156),(69,168,80,157),(70,169,81,158),(71,170,82,159),(72,171,83,160),(73,172,84,161),(74,173,85,162),(75,174,86,163),(76,175,87,164),(77,176,88,165)], [(1,76,32,54),(2,77,33,55),(3,67,23,45),(4,68,24,46),(5,69,25,47),(6,70,26,48),(7,71,27,49),(8,72,28,50),(9,73,29,51),(10,74,30,52),(11,75,31,53),(12,78,34,56),(13,79,35,57),(14,80,36,58),(15,81,37,59),(16,82,38,60),(17,83,39,61),(18,84,40,62),(19,85,41,63),(20,86,42,64),(21,87,43,65),(22,88,44,66),(89,155,111,133),(90,156,112,134),(91,157,113,135),(92,158,114,136),(93,159,115,137),(94,160,116,138),(95,161,117,139),(96,162,118,140),(97,163,119,141),(98,164,120,142),(99,165,121,143),(100,166,122,144),(101,167,123,145),(102,168,124,146),(103,169,125,147),(104,170,126,148),(105,171,127,149),(106,172,128,150),(107,173,129,151),(108,174,130,152),(109,175,131,153),(110,176,132,154)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,66),(53,65),(54,64),(55,63),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,88),(75,87),(76,86),(77,85),(89,95),(90,94),(91,93),(96,99),(97,98),(100,106),(101,105),(102,104),(107,110),(108,109),(111,117),(112,116),(113,115),(118,121),(119,120),(122,128),(123,127),(124,126),(129,132),(130,131),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,154),(141,153),(142,152),(143,151),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,176),(163,175),(164,174),(165,173)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 22 | 22 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | D11 | D22 | C4×D11 | D44⋊5C2 |
| kernel | C42⋊D11 | C4×Dic11 | Dic11⋊C4 | D22⋊C4 | C4×C44 | C2×C4×D11 | C4×D11 | C22 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 4 | 5 | 15 | 20 | 40 |
Matrix representation of C42⋊D11 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 34 | 0 |
| 0 | 0 | 34 |
| 34 | 0 | 0 |
| 0 | 72 | 41 |
| 0 | 6 | 17 |
| 1 | 0 | 0 |
| 0 | 34 | 1 |
| 0 | 11 | 37 |
| 88 | 0 | 0 |
| 0 | 37 | 88 |
| 0 | 33 | 52 |
G:=sub<GL(3,GF(89))| [1,0,0,0,34,0,0,0,34],[34,0,0,0,72,6,0,41,17],[1,0,0,0,34,11,0,1,37],[88,0,0,0,37,33,0,88,52] >;
C42⋊D11 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_{11} % in TeX
G:=Group("C4^2:D11"); // GroupNames label
G:=SmallGroup(352,67);
// by ID
G=gap.SmallGroup(352,67);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,362,50,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations