metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊2D44, C44⋊1D4, D22⋊2D4, C4⋊C4⋊3D11, D22⋊C4⋊8C2, (C2×D44)⋊4C2, C2.9(C2×D44), C22.7(C2×D4), C11⋊2(C4⋊D4), (C2×C4).12D22, C2.13(D4×D11), (C2×C44).5C22, C22.34(C4○D4), (C2×C22).36C23, C2.6(D44⋊C2), (C22×D11).7C22, C22.50(C22×D11), (C2×Dic11).31C22, (C2×C4×D11)⋊1C2, (C11×C4⋊C4)⋊6C2, SmallGroup(352,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊2D44
G = < a,b,c | a4=b44=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 738 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C4⋊D4, Dic11, C44, C44, D22, D22, C2×C22, C4×D11, D44, C2×Dic11, C2×C44, C2×C44, C22×D11, C22×D11, D22⋊C4, C11×C4⋊C4, C2×C4×D11, C2×D44, C2×D44, C4⋊2D44
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4⋊D4, D22, D44, C22×D11, C2×D44, D4×D11, D44⋊C2, C4⋊2D44
►On 176 points
Generators in S176
(1 141 73 126)(2 127 74 142)(3 143 75 128)(4 129 76 144)(5 145 77 130)(6 131 78 146)(7 147 79 132)(8 89 80 148)(9 149 81 90)(10 91 82 150)(11 151 83 92)(12 93 84 152)(13 153 85 94)(14 95 86 154)(15 155 87 96)(16 97 88 156)(17 157 45 98)(18 99 46 158)(19 159 47 100)(20 101 48 160)(21 161 49 102)(22 103 50 162)(23 163 51 104)(24 105 52 164)(25 165 53 106)(26 107 54 166)(27 167 55 108)(28 109 56 168)(29 169 57 110)(30 111 58 170)(31 171 59 112)(32 113 60 172)(33 173 61 114)(34 115 62 174)(35 175 63 116)(36 117 64 176)(37 133 65 118)(38 119 66 134)(39 135 67 120)(40 121 68 136)(41 137 69 122)(42 123 70 138)(43 139 71 124)(44 125 72 140)
(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 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(89 144)(90 143)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 135)(99 134)(100 133)(101 176)(102 175)(103 174)(104 173)(105 172)(106 171)(107 170)(108 169)(109 168)(110 167)(111 166)(112 165)(113 164)(114 163)(115 162)(116 161)(117 160)(118 159)(119 158)(120 157)(121 156)(122 155)(123 154)(124 153)(125 152)(126 151)(127 150)(128 149)(129 148)(130 147)(131 146)(132 145)
G:=sub<Sym(176)| (1,141,73,126)(2,127,74,142)(3,143,75,128)(4,129,76,144)(5,145,77,130)(6,131,78,146)(7,147,79,132)(8,89,80,148)(9,149,81,90)(10,91,82,150)(11,151,83,92)(12,93,84,152)(13,153,85,94)(14,95,86,154)(15,155,87,96)(16,97,88,156)(17,157,45,98)(18,99,46,158)(19,159,47,100)(20,101,48,160)(21,161,49,102)(22,103,50,162)(23,163,51,104)(24,105,52,164)(25,165,53,106)(26,107,54,166)(27,167,55,108)(28,109,56,168)(29,169,57,110)(30,111,58,170)(31,171,59,112)(32,113,60,172)(33,173,61,114)(34,115,62,174)(35,175,63,116)(36,117,64,176)(37,133,65,118)(38,119,66,134)(39,135,67,120)(40,121,68,136)(41,137,69,122)(42,123,70,138)(43,139,71,124)(44,125,72,140), (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)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(89,144)(90,143)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,176)(102,175)(103,174)(104,173)(105,172)(106,171)(107,170)(108,169)(109,168)(110,167)(111,166)(112,165)(113,164)(114,163)(115,162)(116,161)(117,160)(118,159)(119,158)(120,157)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)>;
G:=Group( (1,141,73,126)(2,127,74,142)(3,143,75,128)(4,129,76,144)(5,145,77,130)(6,131,78,146)(7,147,79,132)(8,89,80,148)(9,149,81,90)(10,91,82,150)(11,151,83,92)(12,93,84,152)(13,153,85,94)(14,95,86,154)(15,155,87,96)(16,97,88,156)(17,157,45,98)(18,99,46,158)(19,159,47,100)(20,101,48,160)(21,161,49,102)(22,103,50,162)(23,163,51,104)(24,105,52,164)(25,165,53,106)(26,107,54,166)(27,167,55,108)(28,109,56,168)(29,169,57,110)(30,111,58,170)(31,171,59,112)(32,113,60,172)(33,173,61,114)(34,115,62,174)(35,175,63,116)(36,117,64,176)(37,133,65,118)(38,119,66,134)(39,135,67,120)(40,121,68,136)(41,137,69,122)(42,123,70,138)(43,139,71,124)(44,125,72,140), (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)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(89,144)(90,143)(91,142)(92,141)(93,140)(94,139)(95,138)(96,137)(97,136)(98,135)(99,134)(100,133)(101,176)(102,175)(103,174)(104,173)(105,172)(106,171)(107,170)(108,169)(109,168)(110,167)(111,166)(112,165)(113,164)(114,163)(115,162)(116,161)(117,160)(118,159)(119,158)(120,157)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145) );
G=PermutationGroup([[(1,141,73,126),(2,127,74,142),(3,143,75,128),(4,129,76,144),(5,145,77,130),(6,131,78,146),(7,147,79,132),(8,89,80,148),(9,149,81,90),(10,91,82,150),(11,151,83,92),(12,93,84,152),(13,153,85,94),(14,95,86,154),(15,155,87,96),(16,97,88,156),(17,157,45,98),(18,99,46,158),(19,159,47,100),(20,101,48,160),(21,161,49,102),(22,103,50,162),(23,163,51,104),(24,105,52,164),(25,165,53,106),(26,107,54,166),(27,167,55,108),(28,109,56,168),(29,169,57,110),(30,111,58,170),(31,171,59,112),(32,113,60,172),(33,173,61,114),(34,115,62,174),(35,175,63,116),(36,117,64,176),(37,133,65,118),(38,119,66,134),(39,135,67,120),(40,121,68,136),(41,137,69,122),(42,123,70,138),(43,139,71,124),(44,125,72,140)], [(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),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(89,144),(90,143),(91,142),(92,141),(93,140),(94,139),(95,138),(96,137),(97,136),(98,135),(99,134),(100,133),(101,176),(102,175),(103,174),(104,173),(105,172),(106,171),(107,170),(108,169),(109,168),(110,167),(111,166),(112,165),(113,164),(114,163),(115,162),(116,161),(117,160),(118,159),(119,158),(120,157),(121,156),(122,155),(123,154),(124,153),(125,152),(126,151),(127,150),(128,149),(129,148),(130,147),(131,146),(132,145)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 22 | 22 | 44 | 44 | 2 | 2 | 4 | 4 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D11 | D22 | D44 | D4×D11 | D44⋊C2 |
| kernel | C4⋊2D44 | D22⋊C4 | C11×C4⋊C4 | C2×C4×D11 | C2×D44 | C44 | D22 | C22 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 2 | 5 | 15 | 20 | 5 | 5 |
Matrix representation of C4⋊2D44 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 57 | 34 |
| 67 | 25 | 0 | 0 |
| 8 | 3 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 69 | 88 |
| 31 | 30 | 0 | 0 |
| 57 | 58 | 0 | 0 |
| 0 | 0 | 88 | 80 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,55,57,0,0,0,34],[67,8,0,0,25,3,0,0,0,0,1,69,0,0,9,88],[31,57,0,0,30,58,0,0,0,0,88,0,0,0,80,1] >;
C4⋊2D44 in GAP, Magma, Sage, TeX
C_4\rtimes_2D_{44} % in TeX
G:=Group("C4:2D44"); // GroupNames label
G:=SmallGroup(352,90);
// by ID
G=gap.SmallGroup(352,90);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,188,50,11525]);
// Polycyclic
G:=Group<a,b,c|a^4=b^44=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations