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## G = D4×C2×C22order 352 = 25·11

### Direct product of C2×C22 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C22
 Chief series C1 — C2 — C22 — C2×C22 — D4×C11 — D4×C22 — D4×C2×C22
 Lower central C1 — C2 — D4×C2×C22
 Upper central C1 — C22×C22 — D4×C2×C22

Generators and relations for D4×C2×C22
G = < a,b,c,d | a2=b22=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C23, C11, C22×C4, C2×D4, C24, C22, C22, C22, C22×D4, C44, C2×C22, C2×C22, C2×C44, D4×C11, C22×C22, C22×C22, C22×C22, C22×C44, D4×C22, C23×C22, D4×C2×C22
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C24, C22, C22×D4, C2×C22, D4×C11, C22×C22, D4×C22, C23×C22, D4×C2×C22

Smallest permutation representation of D4×C2×C22
On 176 points
Generators in S176
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 145)(13 146)(14 147)(15 148)(16 149)(17 150)(18 151)(19 152)(20 153)(21 154)(22 133)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 67)(42 68)(43 69)(44 70)(45 108)(46 109)(47 110)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)
(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)
(1 167 28 90)(2 168 29 91)(3 169 30 92)(4 170 31 93)(5 171 32 94)(6 172 33 95)(7 173 34 96)(8 174 35 97)(9 175 36 98)(10 176 37 99)(11 155 38 100)(12 156 39 101)(13 157 40 102)(14 158 41 103)(15 159 42 104)(16 160 43 105)(17 161 44 106)(18 162 23 107)(19 163 24 108)(20 164 25 109)(21 165 26 110)(22 166 27 89)(45 152 131 72)(46 153 132 73)(47 154 111 74)(48 133 112 75)(49 134 113 76)(50 135 114 77)(51 136 115 78)(52 137 116 79)(53 138 117 80)(54 139 118 81)(55 140 119 82)(56 141 120 83)(57 142 121 84)(58 143 122 85)(59 144 123 86)(60 145 124 87)(61 146 125 88)(62 147 126 67)(63 148 127 68)(64 149 128 69)(65 150 129 70)(66 151 130 71)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 89)(23 162)(24 163)(25 164)(26 165)(27 166)(28 167)(29 168)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 155)(39 156)(40 157)(41 158)(42 159)(43 160)(44 161)(45 152)(46 153)(47 154)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 141)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)

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

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

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

220 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A 4B 4C 4D 11A ··· 11J 22A ··· 22BR 22BS ··· 22ET 44A ··· 44AN order 1 2 ··· 2 2 ··· 2 4 4 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

220 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C11 C22 C22 C22 D4 D4×C11 kernel D4×C2×C22 C22×C44 D4×C22 C23×C22 C22×D4 C22×C4 C2×D4 C24 C2×C22 C22 # reps 1 1 12 2 10 10 120 20 4 40

Matrix representation of D4×C2×C22 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 1 0 0 0 0 1
,
 39 0 0 0 0 88 0 0 0 0 87 0 0 0 0 87
,
 1 0 0 0 0 88 0 0 0 0 1 87 0 0 1 88
,
 88 0 0 0 0 1 0 0 0 0 1 87 0 0 0 88
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[39,0,0,0,0,88,0,0,0,0,87,0,0,0,0,87],[1,0,0,0,0,88,0,0,0,0,1,1,0,0,87,88],[88,0,0,0,0,1,0,0,0,0,1,0,0,0,87,88] >;

D4×C2×C22 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_{22}
% in TeX

G:=Group("D4xC2xC22");
// GroupNames label

G:=SmallGroup(352,189);
// by ID

G=gap.SmallGroup(352,189);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-11,-2,2137]);
// Polycyclic

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

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