metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44⋊2D4, D22⋊3D4, C23.8D22, (D4×C22)⋊3C2, (C2×D4)⋊4D11, C4⋊2(C11⋊D4), C11⋊4(C4⋊D4), C44⋊C4⋊14C2, C22.50(C2×D4), (C2×C4).51D22, C2.26(D4×D11), C22.31(C4○D4), (C2×C44).34C22, (C2×C22).53C23, C23.D11⋊11C2, C2.17(D4⋊2D11), (C22×C22).20C22, C22.60(C22×D11), (C2×Dic11).19C22, (C22×D11).26C22, (C2×C4×D11)⋊2C2, (C2×C11⋊D4)⋊5C2, C2.14(C2×C11⋊D4), SmallGroup(352,133)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44⋊2D4
G = < a,b,c | a44=b4=c2=1, bab-1=a-1, cac=a21, cbc=b-1 >
Subgroups: 538 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, D11, C22, C22, C4⋊D4, Dic11, C44, D22, D22, C2×C22, C2×C22, C4×D11, C2×Dic11, C2×Dic11, C11⋊D4, C2×C44, D4×C11, C22×D11, C22×C22, C44⋊C4, C23.D11, C2×C4×D11, C2×C11⋊D4, D4×C22, C44⋊2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4⋊D4, D22, C11⋊D4, C22×D11, D4×D11, D4⋊2D11, C2×C11⋊D4, C44⋊2D4
►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)
(1 108 77 159)(2 107 78 158)(3 106 79 157)(4 105 80 156)(5 104 81 155)(6 103 82 154)(7 102 83 153)(8 101 84 152)(9 100 85 151)(10 99 86 150)(11 98 87 149)(12 97 88 148)(13 96 45 147)(14 95 46 146)(15 94 47 145)(16 93 48 144)(17 92 49 143)(18 91 50 142)(19 90 51 141)(20 89 52 140)(21 132 53 139)(22 131 54 138)(23 130 55 137)(24 129 56 136)(25 128 57 135)(26 127 58 134)(27 126 59 133)(28 125 60 176)(29 124 61 175)(30 123 62 174)(31 122 63 173)(32 121 64 172)(33 120 65 171)(34 119 66 170)(35 118 67 169)(36 117 68 168)(37 116 69 167)(38 115 70 166)(39 114 71 165)(40 113 72 164)(41 112 73 163)(42 111 74 162)(43 110 75 161)(44 109 76 160)
(1 23)(2 44)(3 21)(4 42)(5 19)(6 40)(7 17)(8 38)(9 15)(10 36)(11 13)(12 34)(14 32)(16 30)(18 28)(20 26)(22 24)(25 43)(27 41)(29 39)(31 37)(33 35)(45 87)(46 64)(47 85)(48 62)(49 83)(50 60)(51 81)(52 58)(53 79)(54 56)(55 77)(57 75)(59 73)(61 71)(63 69)(65 67)(66 88)(68 86)(70 84)(72 82)(74 80)(76 78)(89 134)(90 155)(91 176)(92 153)(93 174)(94 151)(95 172)(96 149)(97 170)(98 147)(99 168)(100 145)(101 166)(102 143)(103 164)(104 141)(105 162)(106 139)(107 160)(108 137)(109 158)(110 135)(111 156)(112 133)(113 154)(114 175)(115 152)(116 173)(117 150)(118 171)(119 148)(120 169)(121 146)(122 167)(123 144)(124 165)(125 142)(126 163)(127 140)(128 161)(129 138)(130 159)(131 136)(132 157)
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), (1,108,77,159)(2,107,78,158)(3,106,79,157)(4,105,80,156)(5,104,81,155)(6,103,82,154)(7,102,83,153)(8,101,84,152)(9,100,85,151)(10,99,86,150)(11,98,87,149)(12,97,88,148)(13,96,45,147)(14,95,46,146)(15,94,47,145)(16,93,48,144)(17,92,49,143)(18,91,50,142)(19,90,51,141)(20,89,52,140)(21,132,53,139)(22,131,54,138)(23,130,55,137)(24,129,56,136)(25,128,57,135)(26,127,58,134)(27,126,59,133)(28,125,60,176)(29,124,61,175)(30,123,62,174)(31,122,63,173)(32,121,64,172)(33,120,65,171)(34,119,66,170)(35,118,67,169)(36,117,68,168)(37,116,69,167)(38,115,70,166)(39,114,71,165)(40,113,72,164)(41,112,73,163)(42,111,74,162)(43,110,75,161)(44,109,76,160), (1,23)(2,44)(3,21)(4,42)(5,19)(6,40)(7,17)(8,38)(9,15)(10,36)(11,13)(12,34)(14,32)(16,30)(18,28)(20,26)(22,24)(25,43)(27,41)(29,39)(31,37)(33,35)(45,87)(46,64)(47,85)(48,62)(49,83)(50,60)(51,81)(52,58)(53,79)(54,56)(55,77)(57,75)(59,73)(61,71)(63,69)(65,67)(66,88)(68,86)(70,84)(72,82)(74,80)(76,78)(89,134)(90,155)(91,176)(92,153)(93,174)(94,151)(95,172)(96,149)(97,170)(98,147)(99,168)(100,145)(101,166)(102,143)(103,164)(104,141)(105,162)(106,139)(107,160)(108,137)(109,158)(110,135)(111,156)(112,133)(113,154)(114,175)(115,152)(116,173)(117,150)(118,171)(119,148)(120,169)(121,146)(122,167)(123,144)(124,165)(125,142)(126,163)(127,140)(128,161)(129,138)(130,159)(131,136)(132,157)>;
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), (1,108,77,159)(2,107,78,158)(3,106,79,157)(4,105,80,156)(5,104,81,155)(6,103,82,154)(7,102,83,153)(8,101,84,152)(9,100,85,151)(10,99,86,150)(11,98,87,149)(12,97,88,148)(13,96,45,147)(14,95,46,146)(15,94,47,145)(16,93,48,144)(17,92,49,143)(18,91,50,142)(19,90,51,141)(20,89,52,140)(21,132,53,139)(22,131,54,138)(23,130,55,137)(24,129,56,136)(25,128,57,135)(26,127,58,134)(27,126,59,133)(28,125,60,176)(29,124,61,175)(30,123,62,174)(31,122,63,173)(32,121,64,172)(33,120,65,171)(34,119,66,170)(35,118,67,169)(36,117,68,168)(37,116,69,167)(38,115,70,166)(39,114,71,165)(40,113,72,164)(41,112,73,163)(42,111,74,162)(43,110,75,161)(44,109,76,160), (1,23)(2,44)(3,21)(4,42)(5,19)(6,40)(7,17)(8,38)(9,15)(10,36)(11,13)(12,34)(14,32)(16,30)(18,28)(20,26)(22,24)(25,43)(27,41)(29,39)(31,37)(33,35)(45,87)(46,64)(47,85)(48,62)(49,83)(50,60)(51,81)(52,58)(53,79)(54,56)(55,77)(57,75)(59,73)(61,71)(63,69)(65,67)(66,88)(68,86)(70,84)(72,82)(74,80)(76,78)(89,134)(90,155)(91,176)(92,153)(93,174)(94,151)(95,172)(96,149)(97,170)(98,147)(99,168)(100,145)(101,166)(102,143)(103,164)(104,141)(105,162)(106,139)(107,160)(108,137)(109,158)(110,135)(111,156)(112,133)(113,154)(114,175)(115,152)(116,173)(117,150)(118,171)(119,148)(120,169)(121,146)(122,167)(123,144)(124,165)(125,142)(126,163)(127,140)(128,161)(129,138)(130,159)(131,136)(132,157) );
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)], [(1,108,77,159),(2,107,78,158),(3,106,79,157),(4,105,80,156),(5,104,81,155),(6,103,82,154),(7,102,83,153),(8,101,84,152),(9,100,85,151),(10,99,86,150),(11,98,87,149),(12,97,88,148),(13,96,45,147),(14,95,46,146),(15,94,47,145),(16,93,48,144),(17,92,49,143),(18,91,50,142),(19,90,51,141),(20,89,52,140),(21,132,53,139),(22,131,54,138),(23,130,55,137),(24,129,56,136),(25,128,57,135),(26,127,58,134),(27,126,59,133),(28,125,60,176),(29,124,61,175),(30,123,62,174),(31,122,63,173),(32,121,64,172),(33,120,65,171),(34,119,66,170),(35,118,67,169),(36,117,68,168),(37,116,69,167),(38,115,70,166),(39,114,71,165),(40,113,72,164),(41,112,73,163),(42,111,74,162),(43,110,75,161),(44,109,76,160)], [(1,23),(2,44),(3,21),(4,42),(5,19),(6,40),(7,17),(8,38),(9,15),(10,36),(11,13),(12,34),(14,32),(16,30),(18,28),(20,26),(22,24),(25,43),(27,41),(29,39),(31,37),(33,35),(45,87),(46,64),(47,85),(48,62),(49,83),(50,60),(51,81),(52,58),(53,79),(54,56),(55,77),(57,75),(59,73),(61,71),(63,69),(65,67),(66,88),(68,86),(70,84),(72,82),(74,80),(76,78),(89,134),(90,155),(91,176),(92,153),(93,174),(94,151),(95,172),(96,149),(97,170),(98,147),(99,168),(100,145),(101,166),(102,143),(103,164),(104,141),(105,162),(106,139),(107,160),(108,137),(109,158),(110,135),(111,156),(112,133),(113,154),(114,175),(115,152),(116,173),(117,150),(118,171),(119,148),(120,169),(121,146),(122,167),(123,144),(124,165),(125,142),(126,163),(127,140),(128,161),(129,138),(130,159),(131,136),(132,157)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22AI | 44A | ··· | 44J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 22 | 22 | 2 | 2 | 22 | 22 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D11 | D22 | D22 | C11⋊D4 | D4×D11 | D4⋊2D11 |
| kernel | C44⋊2D4 | C44⋊C4 | C23.D11 | C2×C4×D11 | C2×C11⋊D4 | D4×C22 | C44 | D22 | C22 | C2×D4 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 20 | 5 | 5 |
Matrix representation of C44⋊2D4 ►in GL6(𝔽89)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 8 | 0 | 0 |
| 0 | 0 | 83 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 0 | 77 | 55 |
| 48 | 53 | 0 | 0 | 0 | 0 |
| 22 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 3 | 0 | 0 |
| 0 | 0 | 54 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 71 |
| 0 | 0 | 0 | 0 | 69 | 66 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 62 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 3 | 0 | 0 |
| 0 | 0 | 54 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 0 | 37 | 1 |
G:=sub<GL(6,GF(89))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,58,83,0,0,0,0,8,13,0,0,0,0,0,0,34,77,0,0,0,0,0,55],[48,22,0,0,0,0,53,41,0,0,0,0,0,0,62,54,0,0,0,0,3,27,0,0,0,0,0,0,23,69,0,0,0,0,71,66],[1,62,0,0,0,0,0,88,0,0,0,0,0,0,62,54,0,0,0,0,3,27,0,0,0,0,0,0,88,37,0,0,0,0,0,1] >;
C44⋊2D4 in GAP, Magma, Sage, TeX
C_{44}\rtimes_2D_4 % in TeX
G:=Group("C44:2D4"); // GroupNames label
G:=SmallGroup(352,133);
// by ID
G=gap.SmallGroup(352,133);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,103,218,188,11525]);
// Polycyclic
G:=Group<a,b,c|a^44=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^21,c*b*c=b^-1>;
// generators/relations