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## G = D4×C44order 352 = 25·11

### Direct product of C44 and D4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C44
G = < a,b,c | a44=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C11, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C22, C22, C4×D4, C44, C44, C2×C22, C2×C22, C2×C22, C2×C44, C2×C44, C2×C44, D4×C11, C22×C22, C4×C44, C11×C22⋊C4, C11×C4⋊C4, C22×C44, D4×C22, D4×C44
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C11, C22×C4, C2×D4, C4○D4, C22, C4×D4, C44, C2×C22, C2×C44, D4×C11, C22×C22, C22×C44, D4×C22, C11×C4○D4, D4×C44

Smallest permutation representation of D4×C44
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)
(1 171 74 122)(2 172 75 123)(3 173 76 124)(4 174 77 125)(5 175 78 126)(6 176 79 127)(7 133 80 128)(8 134 81 129)(9 135 82 130)(10 136 83 131)(11 137 84 132)(12 138 85 89)(13 139 86 90)(14 140 87 91)(15 141 88 92)(16 142 45 93)(17 143 46 94)(18 144 47 95)(19 145 48 96)(20 146 49 97)(21 147 50 98)(22 148 51 99)(23 149 52 100)(24 150 53 101)(25 151 54 102)(26 152 55 103)(27 153 56 104)(28 154 57 105)(29 155 58 106)(30 156 59 107)(31 157 60 108)(32 158 61 109)(33 159 62 110)(34 160 63 111)(35 161 64 112)(36 162 65 113)(37 163 66 114)(38 164 67 115)(39 165 68 116)(40 166 69 117)(41 167 70 118)(42 168 71 119)(43 169 72 120)(44 170 73 121)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(89 160)(90 161)(91 162)(92 163)(93 164)(94 165)(95 166)(96 167)(97 168)(98 169)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(105 176)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(118 145)(119 146)(120 147)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)

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

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

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

220 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4L 11A ··· 11J 22A ··· 22AD 22AE ··· 22BR 44A ··· 44AN 44AO ··· 44DP order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 size 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

220 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C11 C22 C22 C22 C22 C22 C44 D4 C4○D4 D4×C11 C11×C4○D4 kernel D4×C44 C4×C44 C11×C22⋊C4 C11×C4⋊C4 C22×C44 D4×C22 D4×C11 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C44 C22 C4 C2 # reps 1 1 2 1 2 1 8 10 10 20 10 20 10 80 2 2 20 20

Matrix representation of D4×C44 in GL3(𝔽89) generated by

 42 0 0 0 34 0 0 0 34
,
 1 0 0 0 54 2 0 10 35
,
 1 0 0 0 88 0 0 54 1
G:=sub<GL(3,GF(89))| [42,0,0,0,34,0,0,0,34],[1,0,0,0,54,10,0,2,35],[1,0,0,0,88,54,0,0,1] >;

D4×C44 in GAP, Magma, Sage, TeX

D_4\times C_{44}
% in TeX

G:=Group("D4xC44");
// GroupNames label

G:=SmallGroup(352,153);
// by ID

G=gap.SmallGroup(352,153);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1056,1081,806]);
// Polycyclic

G:=Group<a,b,c|a^44=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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