metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊1D44, C44⋊4D4, C42⋊5D11, (C4×C44)⋊4C2, (C2×D44)⋊1C2, C22.3(C2×D4), C2.5(C2×D44), C11⋊1(C4⋊1D4), (C2×C4).76D22, (C2×C22).15C23, (C2×C44).87C22, (C22×D11).1C22, C22.36(C22×D11), SmallGroup(352,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D44
G = < a,b,c | a4=b44=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 930 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C11, C42, C2×D4, D11, C22, C4⋊1D4, C44, D22, C2×C22, D44, C2×C44, C22×D11, C4×C44, C2×D44, C4⋊D44
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C4⋊1D4, D22, D44, C22×D11, C2×D44, C4⋊D44
►On 176 points
Generators in S176
(1 162 117 72)(2 163 118 73)(3 164 119 74)(4 165 120 75)(5 166 121 76)(6 167 122 77)(7 168 123 78)(8 169 124 79)(9 170 125 80)(10 171 126 81)(11 172 127 82)(12 173 128 83)(13 174 129 84)(14 175 130 85)(15 176 131 86)(16 133 132 87)(17 134 89 88)(18 135 90 45)(19 136 91 46)(20 137 92 47)(21 138 93 48)(22 139 94 49)(23 140 95 50)(24 141 96 51)(25 142 97 52)(26 143 98 53)(27 144 99 54)(28 145 100 55)(29 146 101 56)(30 147 102 57)(31 148 103 58)(32 149 104 59)(33 150 105 60)(34 151 106 61)(35 152 107 62)(36 153 108 63)(37 154 109 64)(38 155 110 65)(39 156 111 66)(40 157 112 67)(41 158 113 68)(42 159 114 69)(43 160 115 70)(44 161 116 71)
(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 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 148)(53 147)(54 146)(55 145)(56 144)(57 143)(58 142)(59 141)(60 140)(61 139)(62 138)(63 137)(64 136)(65 135)(66 134)(67 133)(68 176)(69 175)(70 174)(71 173)(72 172)(73 171)(74 170)(75 169)(76 168)(77 167)(78 166)(79 165)(80 164)(81 163)(82 162)(83 161)(84 160)(85 159)(86 158)(87 157)(88 156)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)
G:=sub<Sym(176)| (1,162,117,72)(2,163,118,73)(3,164,119,74)(4,165,120,75)(5,166,121,76)(6,167,122,77)(7,168,123,78)(8,169,124,79)(9,170,125,80)(10,171,126,81)(11,172,127,82)(12,173,128,83)(13,174,129,84)(14,175,130,85)(15,176,131,86)(16,133,132,87)(17,134,89,88)(18,135,90,45)(19,136,91,46)(20,137,92,47)(21,138,93,48)(22,139,94,49)(23,140,95,50)(24,141,96,51)(25,142,97,52)(26,143,98,53)(27,144,99,54)(28,145,100,55)(29,146,101,56)(30,147,102,57)(31,148,103,58)(32,149,104,59)(33,150,105,60)(34,151,106,61)(35,152,107,62)(36,153,108,63)(37,154,109,64)(38,155,110,65)(39,156,111,66)(40,157,112,67)(41,158,113,68)(42,159,114,69)(43,160,115,70)(44,161,116,71), (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)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,147)(54,146)(55,145)(56,144)(57,143)(58,142)(59,141)(60,140)(61,139)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,176)(69,175)(70,174)(71,173)(72,172)(73,171)(74,170)(75,169)(76,168)(77,167)(78,166)(79,165)(80,164)(81,163)(82,162)(83,161)(84,160)(85,159)(86,158)(87,157)(88,156)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(112,132)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123)>;
G:=Group( (1,162,117,72)(2,163,118,73)(3,164,119,74)(4,165,120,75)(5,166,121,76)(6,167,122,77)(7,168,123,78)(8,169,124,79)(9,170,125,80)(10,171,126,81)(11,172,127,82)(12,173,128,83)(13,174,129,84)(14,175,130,85)(15,176,131,86)(16,133,132,87)(17,134,89,88)(18,135,90,45)(19,136,91,46)(20,137,92,47)(21,138,93,48)(22,139,94,49)(23,140,95,50)(24,141,96,51)(25,142,97,52)(26,143,98,53)(27,144,99,54)(28,145,100,55)(29,146,101,56)(30,147,102,57)(31,148,103,58)(32,149,104,59)(33,150,105,60)(34,151,106,61)(35,152,107,62)(36,153,108,63)(37,154,109,64)(38,155,110,65)(39,156,111,66)(40,157,112,67)(41,158,113,68)(42,159,114,69)(43,160,115,70)(44,161,116,71), (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)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,147)(54,146)(55,145)(56,144)(57,143)(58,142)(59,141)(60,140)(61,139)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,176)(69,175)(70,174)(71,173)(72,172)(73,171)(74,170)(75,169)(76,168)(77,167)(78,166)(79,165)(80,164)(81,163)(82,162)(83,161)(84,160)(85,159)(86,158)(87,157)(88,156)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(112,132)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)(121,123) );
G=PermutationGroup([[(1,162,117,72),(2,163,118,73),(3,164,119,74),(4,165,120,75),(5,166,121,76),(6,167,122,77),(7,168,123,78),(8,169,124,79),(9,170,125,80),(10,171,126,81),(11,172,127,82),(12,173,128,83),(13,174,129,84),(14,175,130,85),(15,176,131,86),(16,133,132,87),(17,134,89,88),(18,135,90,45),(19,136,91,46),(20,137,92,47),(21,138,93,48),(22,139,94,49),(23,140,95,50),(24,141,96,51),(25,142,97,52),(26,143,98,53),(27,144,99,54),(28,145,100,55),(29,146,101,56),(30,147,102,57),(31,148,103,58),(32,149,104,59),(33,150,105,60),(34,151,106,61),(35,152,107,62),(36,153,108,63),(37,154,109,64),(38,155,110,65),(39,156,111,66),(40,157,112,67),(41,158,113,68),(42,159,114,69),(43,160,115,70),(44,161,116,71)], [(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),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(45,155),(46,154),(47,153),(48,152),(49,151),(50,150),(51,149),(52,148),(53,147),(54,146),(55,145),(56,144),(57,143),(58,142),(59,141),(60,140),(61,139),(62,138),(63,137),(64,136),(65,135),(66,134),(67,133),(68,176),(69,175),(70,174),(71,173),(72,172),(73,171),(74,170),(75,169),(76,168),(77,167),(78,166),(79,165),(80,164),(81,163),(82,162),(83,161),(84,160),(85,159),(86,158),(87,157),(88,156),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(112,132),(113,131),(114,130),(115,129),(116,128),(117,127),(118,126),(119,125),(120,124),(121,123)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 44 | 44 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D11 | D22 | D44 |
| kernel | C4⋊D44 | C4×C44 | C2×D44 | C44 | C42 | C2×C4 | C4 |
| # reps | 1 | 1 | 6 | 6 | 5 | 15 | 60 |
Matrix representation of C4⋊D44 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 61 |
| 0 | 0 | 70 | 1 |
| 80 | 35 | 0 | 0 |
| 54 | 47 | 0 | 0 |
| 0 | 0 | 49 | 2 |
| 0 | 0 | 84 | 87 |
| 76 | 87 | 0 | 0 |
| 84 | 13 | 0 | 0 |
| 0 | 0 | 83 | 84 |
| 0 | 0 | 7 | 6 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,70,0,0,61,1],[80,54,0,0,35,47,0,0,0,0,49,84,0,0,2,87],[76,84,0,0,87,13,0,0,0,0,83,7,0,0,84,6] >;
C4⋊D44 in GAP, Magma, Sage, TeX
C_4\rtimes D_{44} % in TeX
G:=Group("C4:D44"); // GroupNames label
G:=SmallGroup(352,69);
// by ID
G=gap.SmallGroup(352,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,218,50,11525]);
// Polycyclic
G:=Group<a,b,c|a^4=b^44=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations