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## G = D4⋊Dic11order 352 = 25·11

### 1st semidirect product of D4 and Dic11 acting via Dic11/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — D4⋊Dic11
 Chief series C1 — C11 — C22 — C2×C22 — C2×C44 — C44⋊C4 — D4⋊Dic11
 Lower central C11 — C22 — C44 — D4⋊Dic11
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for D4⋊Dic11
G = < a,b,c,d | a4=b2=c22=1, d2=c11, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Smallest permutation representation of D4⋊Dic11
On 176 points
Generators in S176
(1 81 60 99)(2 82 61 100)(3 83 62 101)(4 84 63 102)(5 85 64 103)(6 86 65 104)(7 87 66 105)(8 88 45 106)(9 67 46 107)(10 68 47 108)(11 69 48 109)(12 70 49 110)(13 71 50 89)(14 72 51 90)(15 73 52 91)(16 74 53 92)(17 75 54 93)(18 76 55 94)(19 77 56 95)(20 78 57 96)(21 79 58 97)(22 80 59 98)(23 149 174 129)(24 150 175 130)(25 151 176 131)(26 152 155 132)(27 153 156 111)(28 154 157 112)(29 133 158 113)(30 134 159 114)(31 135 160 115)(32 136 161 116)(33 137 162 117)(34 138 163 118)(35 139 164 119)(36 140 165 120)(37 141 166 121)(38 142 167 122)(39 143 168 123)(40 144 169 124)(41 145 170 125)(42 146 171 126)(43 147 172 127)(44 148 173 128)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(45 88)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(111 153)(112 154)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)
(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)
(1 152 12 141)(2 151 13 140)(3 150 14 139)(4 149 15 138)(5 148 16 137)(6 147 17 136)(7 146 18 135)(8 145 19 134)(9 144 20 133)(10 143 21 154)(11 142 22 153)(23 73 34 84)(24 72 35 83)(25 71 36 82)(26 70 37 81)(27 69 38 80)(28 68 39 79)(29 67 40 78)(30 88 41 77)(31 87 42 76)(32 86 43 75)(33 85 44 74)(45 125 56 114)(46 124 57 113)(47 123 58 112)(48 122 59 111)(49 121 60 132)(50 120 61 131)(51 119 62 130)(52 118 63 129)(53 117 64 128)(54 116 65 127)(55 115 66 126)(89 165 100 176)(90 164 101 175)(91 163 102 174)(92 162 103 173)(93 161 104 172)(94 160 105 171)(95 159 106 170)(96 158 107 169)(97 157 108 168)(98 156 109 167)(99 155 110 166)

G:=sub<Sym(176)| (1,81,60,99)(2,82,61,100)(3,83,62,101)(4,84,63,102)(5,85,64,103)(6,86,65,104)(7,87,66,105)(8,88,45,106)(9,67,46,107)(10,68,47,108)(11,69,48,109)(12,70,49,110)(13,71,50,89)(14,72,51,90)(15,73,52,91)(16,74,53,92)(17,75,54,93)(18,76,55,94)(19,77,56,95)(20,78,57,96)(21,79,58,97)(22,80,59,98)(23,149,174,129)(24,150,175,130)(25,151,176,131)(26,152,155,132)(27,153,156,111)(28,154,157,112)(29,133,158,113)(30,134,159,114)(31,135,160,115)(32,136,161,116)(33,137,162,117)(34,138,163,118)(35,139,164,119)(36,140,165,120)(37,141,166,121)(38,142,167,122)(39,143,168,123)(40,144,169,124)(41,145,170,125)(42,146,171,126)(43,147,172,127)(44,148,173,128), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(45,88)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(111,153)(112,154)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152), (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), (1,152,12,141)(2,151,13,140)(3,150,14,139)(4,149,15,138)(5,148,16,137)(6,147,17,136)(7,146,18,135)(8,145,19,134)(9,144,20,133)(10,143,21,154)(11,142,22,153)(23,73,34,84)(24,72,35,83)(25,71,36,82)(26,70,37,81)(27,69,38,80)(28,68,39,79)(29,67,40,78)(30,88,41,77)(31,87,42,76)(32,86,43,75)(33,85,44,74)(45,125,56,114)(46,124,57,113)(47,123,58,112)(48,122,59,111)(49,121,60,132)(50,120,61,131)(51,119,62,130)(52,118,63,129)(53,117,64,128)(54,116,65,127)(55,115,66,126)(89,165,100,176)(90,164,101,175)(91,163,102,174)(92,162,103,173)(93,161,104,172)(94,160,105,171)(95,159,106,170)(96,158,107,169)(97,157,108,168)(98,156,109,167)(99,155,110,166)>;

G:=Group( (1,81,60,99)(2,82,61,100)(3,83,62,101)(4,84,63,102)(5,85,64,103)(6,86,65,104)(7,87,66,105)(8,88,45,106)(9,67,46,107)(10,68,47,108)(11,69,48,109)(12,70,49,110)(13,71,50,89)(14,72,51,90)(15,73,52,91)(16,74,53,92)(17,75,54,93)(18,76,55,94)(19,77,56,95)(20,78,57,96)(21,79,58,97)(22,80,59,98)(23,149,174,129)(24,150,175,130)(25,151,176,131)(26,152,155,132)(27,153,156,111)(28,154,157,112)(29,133,158,113)(30,134,159,114)(31,135,160,115)(32,136,161,116)(33,137,162,117)(34,138,163,118)(35,139,164,119)(36,140,165,120)(37,141,166,121)(38,142,167,122)(39,143,168,123)(40,144,169,124)(41,145,170,125)(42,146,171,126)(43,147,172,127)(44,148,173,128), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,109)(12,110)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(45,88)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(111,153)(112,154)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152), (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), (1,152,12,141)(2,151,13,140)(3,150,14,139)(4,149,15,138)(5,148,16,137)(6,147,17,136)(7,146,18,135)(8,145,19,134)(9,144,20,133)(10,143,21,154)(11,142,22,153)(23,73,34,84)(24,72,35,83)(25,71,36,82)(26,70,37,81)(27,69,38,80)(28,68,39,79)(29,67,40,78)(30,88,41,77)(31,87,42,76)(32,86,43,75)(33,85,44,74)(45,125,56,114)(46,124,57,113)(47,123,58,112)(48,122,59,111)(49,121,60,132)(50,120,61,131)(51,119,62,130)(52,118,63,129)(53,117,64,128)(54,116,65,127)(55,115,66,126)(89,165,100,176)(90,164,101,175)(91,163,102,174)(92,162,103,173)(93,161,104,172)(94,160,105,171)(95,159,106,170)(96,158,107,169)(97,157,108,168)(98,156,109,167)(99,155,110,166) );

G=PermutationGroup([[(1,81,60,99),(2,82,61,100),(3,83,62,101),(4,84,63,102),(5,85,64,103),(6,86,65,104),(7,87,66,105),(8,88,45,106),(9,67,46,107),(10,68,47,108),(11,69,48,109),(12,70,49,110),(13,71,50,89),(14,72,51,90),(15,73,52,91),(16,74,53,92),(17,75,54,93),(18,76,55,94),(19,77,56,95),(20,78,57,96),(21,79,58,97),(22,80,59,98),(23,149,174,129),(24,150,175,130),(25,151,176,131),(26,152,155,132),(27,153,156,111),(28,154,157,112),(29,133,158,113),(30,134,159,114),(31,135,160,115),(32,136,161,116),(33,137,162,117),(34,138,163,118),(35,139,164,119),(36,140,165,120),(37,141,166,121),(38,142,167,122),(39,143,168,123),(40,144,169,124),(41,145,170,125),(42,146,171,126),(43,147,172,127),(44,148,173,128)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,105),(8,106),(9,107),(10,108),(11,109),(12,110),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,98),(45,88),(46,67),(47,68),(48,69),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(111,153),(112,154),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152)], [(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)], [(1,152,12,141),(2,151,13,140),(3,150,14,139),(4,149,15,138),(5,148,16,137),(6,147,17,136),(7,146,18,135),(8,145,19,134),(9,144,20,133),(10,143,21,154),(11,142,22,153),(23,73,34,84),(24,72,35,83),(25,71,36,82),(26,70,37,81),(27,69,38,80),(28,68,39,79),(29,67,40,78),(30,88,41,77),(31,87,42,76),(32,86,43,75),(33,85,44,74),(45,125,56,114),(46,124,57,113),(47,123,58,112),(48,122,59,111),(49,121,60,132),(50,120,61,131),(51,119,62,130),(52,118,63,129),(53,117,64,128),(54,116,65,127),(55,115,66,126),(89,165,100,176),(90,164,101,175),(91,163,102,174),(92,162,103,173),(93,161,104,172),(94,160,105,171),(95,159,106,170),(96,158,107,169),(97,157,108,168),(98,156,109,167),(99,155,110,166)]])

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A 8B 8C 8D 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J order 1 2 2 2 2 2 4 4 4 4 8 8 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 1 1 4 4 2 2 44 44 22 22 22 22 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C4 D4 D4 D8 SD16 D11 D22 Dic11 C11⋊D4 C11⋊D4 D4⋊D11 D4.D11 kernel D4⋊Dic11 C2×C11⋊C8 C44⋊C4 D4×C22 D4×C11 C44 C2×C22 C22 C22 C2×D4 C2×C4 D4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 5 5 10 10 10 5 5

Matrix representation of D4⋊Dic11 in GL4(𝔽89) generated by

 1 0 0 0 0 1 0 0 0 0 1 5 0 0 53 88
,
 1 0 0 0 0 1 0 0 0 0 1 5 0 0 0 88
,
 44 0 0 0 0 87 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 88 0 0 0 0 0 0 18 0 0 5 0
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,1,53,0,0,5,88],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,5,88],[44,0,0,0,0,87,0,0,0,0,1,0,0,0,0,1],[0,88,0,0,1,0,0,0,0,0,0,5,0,0,18,0] >;

D4⋊Dic11 in GAP, Magma, Sage, TeX

D_4\rtimes {\rm Dic}_{11}
% in TeX

G:=Group("D4:Dic11");
// GroupNames label

G:=SmallGroup(352,38);
// by ID

G=gap.SmallGroup(352,38);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,579,297,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^22=1,d^2=c^11,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

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