metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D22⋊C8, C44.52D4, C4.19D44, C22.3M4(2), (C2×C88)⋊1C2, (C2×C8)⋊1D11, C22.5(C2×C8), C2.5(C8×D11), C11⋊1(C22⋊C8), (C2×C4).93D22, C2.1(D22⋊C4), C2.3(C88⋊C2), C4.27(C11⋊D4), C22.6(C22⋊C4), (C2×Dic11).4C4, (C22×D11).2C4, C22.11(C4×D11), (C2×C44).107C22, (C2×C11⋊C8)⋊9C2, (C2×C4×D11).7C2, (C2×C22).12(C2×C4), SmallGroup(352,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D22⋊C8
G = < a,b,c | a22=b2=c8=1, bab=a-1, ac=ca, cbc-1=a11b >
22C2
22C2
11C22
11C22
22C22
22C22
22C4
2D11
2D11
2C8
11C2×C4
11C23
22C8
22C2×C4
22C2×C4
2D22
2D22
11C2×C8
11C22×C4
2C88
11C22⋊C8
Smallest permutation representation of D22⋊C8
►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)
(1 148)(2 147)(3 146)(4 145)(5 144)(6 143)(7 142)(8 141)(9 140)(10 139)(11 138)(12 137)(13 136)(14 135)(15 134)(16 133)(17 154)(18 153)(19 152)(20 151)(21 150)(22 149)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 90)(30 89)(31 110)(32 109)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 100)(42 99)(43 98)(44 97)(45 88)(46 87)(47 86)(48 85)(49 84)(50 83)(51 82)(52 81)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(111 163)(112 162)(113 161)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 176)(121 175)(122 174)(123 173)(124 172)(125 171)(126 170)(127 169)(128 168)(129 167)(130 166)(131 165)(132 164)
(1 44 77 120 149 109 57 166)(2 23 78 121 150 110 58 167)(3 24 79 122 151 89 59 168)(4 25 80 123 152 90 60 169)(5 26 81 124 153 91 61 170)(6 27 82 125 154 92 62 171)(7 28 83 126 133 93 63 172)(8 29 84 127 134 94 64 173)(9 30 85 128 135 95 65 174)(10 31 86 129 136 96 66 175)(11 32 87 130 137 97 45 176)(12 33 88 131 138 98 46 155)(13 34 67 132 139 99 47 156)(14 35 68 111 140 100 48 157)(15 36 69 112 141 101 49 158)(16 37 70 113 142 102 50 159)(17 38 71 114 143 103 51 160)(18 39 72 115 144 104 52 161)(19 40 73 116 145 105 53 162)(20 41 74 117 146 106 54 163)(21 42 75 118 147 107 55 164)(22 43 76 119 148 108 56 165)
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), (1,148)(2,147)(3,146)(4,145)(5,144)(6,143)(7,142)(8,141)(9,140)(10,139)(11,138)(12,137)(13,136)(14,135)(15,134)(16,133)(17,154)(18,153)(19,152)(20,151)(21,150)(22,149)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,176)(121,175)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,165)(132,164), (1,44,77,120,149,109,57,166)(2,23,78,121,150,110,58,167)(3,24,79,122,151,89,59,168)(4,25,80,123,152,90,60,169)(5,26,81,124,153,91,61,170)(6,27,82,125,154,92,62,171)(7,28,83,126,133,93,63,172)(8,29,84,127,134,94,64,173)(9,30,85,128,135,95,65,174)(10,31,86,129,136,96,66,175)(11,32,87,130,137,97,45,176)(12,33,88,131,138,98,46,155)(13,34,67,132,139,99,47,156)(14,35,68,111,140,100,48,157)(15,36,69,112,141,101,49,158)(16,37,70,113,142,102,50,159)(17,38,71,114,143,103,51,160)(18,39,72,115,144,104,52,161)(19,40,73,116,145,105,53,162)(20,41,74,117,146,106,54,163)(21,42,75,118,147,107,55,164)(22,43,76,119,148,108,56,165)>;
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), (1,148)(2,147)(3,146)(4,145)(5,144)(6,143)(7,142)(8,141)(9,140)(10,139)(11,138)(12,137)(13,136)(14,135)(15,134)(16,133)(17,154)(18,153)(19,152)(20,151)(21,150)(22,149)(23,96)(24,95)(25,94)(26,93)(27,92)(28,91)(29,90)(30,89)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,176)(121,175)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,165)(132,164), (1,44,77,120,149,109,57,166)(2,23,78,121,150,110,58,167)(3,24,79,122,151,89,59,168)(4,25,80,123,152,90,60,169)(5,26,81,124,153,91,61,170)(6,27,82,125,154,92,62,171)(7,28,83,126,133,93,63,172)(8,29,84,127,134,94,64,173)(9,30,85,128,135,95,65,174)(10,31,86,129,136,96,66,175)(11,32,87,130,137,97,45,176)(12,33,88,131,138,98,46,155)(13,34,67,132,139,99,47,156)(14,35,68,111,140,100,48,157)(15,36,69,112,141,101,49,158)(16,37,70,113,142,102,50,159)(17,38,71,114,143,103,51,160)(18,39,72,115,144,104,52,161)(19,40,73,116,145,105,53,162)(20,41,74,117,146,106,54,163)(21,42,75,118,147,107,55,164)(22,43,76,119,148,108,56,165) );
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)], [(1,148),(2,147),(3,146),(4,145),(5,144),(6,143),(7,142),(8,141),(9,140),(10,139),(11,138),(12,137),(13,136),(14,135),(15,134),(16,133),(17,154),(18,153),(19,152),(20,151),(21,150),(22,149),(23,96),(24,95),(25,94),(26,93),(27,92),(28,91),(29,90),(30,89),(31,110),(32,109),(33,108),(34,107),(35,106),(36,105),(37,104),(38,103),(39,102),(40,101),(41,100),(42,99),(43,98),(44,97),(45,88),(46,87),(47,86),(48,85),(49,84),(50,83),(51,82),(52,81),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(111,163),(112,162),(113,161),(114,160),(115,159),(116,158),(117,157),(118,156),(119,155),(120,176),(121,175),(122,174),(123,173),(124,172),(125,171),(126,170),(127,169),(128,168),(129,167),(130,166),(131,165),(132,164)], [(1,44,77,120,149,109,57,166),(2,23,78,121,150,110,58,167),(3,24,79,122,151,89,59,168),(4,25,80,123,152,90,60,169),(5,26,81,124,153,91,61,170),(6,27,82,125,154,92,62,171),(7,28,83,126,133,93,63,172),(8,29,84,127,134,94,64,173),(9,30,85,128,135,95,65,174),(10,31,86,129,136,96,66,175),(11,32,87,130,137,97,45,176),(12,33,88,131,138,98,46,155),(13,34,67,132,139,99,47,156),(14,35,68,111,140,100,48,157),(15,36,69,112,141,101,49,158),(16,37,70,113,142,102,50,159),(17,38,71,114,143,103,51,160),(18,39,72,115,144,104,52,161),(19,40,73,116,145,105,53,162),(20,41,74,117,146,106,54,163),(21,42,75,118,147,107,55,164),(22,43,76,119,148,108,56,165)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 22 | 22 | 1 | 1 | 1 | 1 | 22 | 22 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | D11 | D22 | D44 | C11⋊D4 | C4×D11 | C8×D11 | C88⋊C2 |
| kernel | D22⋊C8 | C2×C11⋊C8 | C2×C88 | C2×C4×D11 | C2×Dic11 | C22×D11 | D22 | C44 | C22 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 20 | 20 |
Matrix representation of D22⋊C8 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 44 | 38 |
| 0 | 51 | 38 |
| 1 | 0 | 0 |
| 0 | 51 | 45 |
| 0 | 51 | 38 |
| 37 | 0 | 0 |
| 0 | 20 | 16 |
| 0 | 73 | 69 |
G:=sub<GL(3,GF(89))| [1,0,0,0,44,51,0,38,38],[1,0,0,0,51,51,0,45,38],[37,0,0,0,20,73,0,16,69] >;
D22⋊C8 in GAP, Magma, Sage, TeX
D_{22}\rtimes C_8 % in TeX
G:=Group("D22:C8"); // GroupNames label
G:=SmallGroup(352,26);
// by ID
G=gap.SmallGroup(352,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,121,31,86,11525]);
// Polycyclic
G:=Group<a,b,c|a^22=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^11*b>;
// generators/relations
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