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## G = C4×D44order 352 = 25·11

### Direct product of C4 and D44

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C4×D44
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×D44 — C4×D44
 Lower central C11 — C22 — C4×D44
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×D44
G = < a,b,c | a4=b44=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 594 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C11, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C4×D4, Dic11, C44, C44, D22, D22, C2×C22, C4×D11, D44, C2×Dic11, C2×C44, C22×D11, C44⋊C4, D22⋊C4, C4×C44, C2×C4×D11, C2×D44, C4×D44
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D11, C4×D4, D22, C4×D11, D44, C22×D11, C2×C4×D11, C2×D44, D445C2, C4×D44

Smallest permutation representation of C4×D44
On 176 points
Generators in S176
(1 134 103 59)(2 135 104 60)(3 136 105 61)(4 137 106 62)(5 138 107 63)(6 139 108 64)(7 140 109 65)(8 141 110 66)(9 142 111 67)(10 143 112 68)(11 144 113 69)(12 145 114 70)(13 146 115 71)(14 147 116 72)(15 148 117 73)(16 149 118 74)(17 150 119 75)(18 151 120 76)(19 152 121 77)(20 153 122 78)(21 154 123 79)(22 155 124 80)(23 156 125 81)(24 157 126 82)(25 158 127 83)(26 159 128 84)(27 160 129 85)(28 161 130 86)(29 162 131 87)(30 163 132 88)(31 164 89 45)(32 165 90 46)(33 166 91 47)(34 167 92 48)(35 168 93 49)(36 169 94 50)(37 170 95 51)(38 171 96 52)(39 172 97 53)(40 173 98 54)(41 174 99 55)(42 175 100 56)(43 176 101 57)(44 133 102 58)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(84 88)(85 87)(89 127)(90 126)(91 125)(92 124)(93 123)(94 122)(95 121)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(128 132)(129 131)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)(146 176)(147 175)(148 174)(149 173)(150 172)(151 171)(152 170)(153 169)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 11A ··· 11E 22A ··· 22O 44A ··· 44BH order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 22 22 22 22 1 1 1 1 2 2 2 2 22 22 22 22 2 ··· 2 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 D11 D22 C4×D11 D44 D44⋊5C2 kernel C4×D44 C44⋊C4 D22⋊C4 C4×C44 C2×C4×D11 C2×D44 D44 C44 C22 C42 C2×C4 C4 C4 C2 # reps 1 1 2 1 2 1 8 2 2 5 15 20 20 20

Matrix representation of C4×D44 in GL3(𝔽89) generated by

 55 0 0 0 34 0 0 0 34
,
 88 0 0 0 52 35 0 58 75
,
 88 0 0 0 72 5 0 67 17
G:=sub<GL(3,GF(89))| [55,0,0,0,34,0,0,0,34],[88,0,0,0,52,58,0,35,75],[88,0,0,0,72,67,0,5,17] >;

C4×D44 in GAP, Magma, Sage, TeX

C_4\times D_{44}
% in TeX

G:=Group("C4xD44");
// GroupNames label

G:=SmallGroup(352,68);
// by ID

G=gap.SmallGroup(352,68);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,103,50,11525]);
// Polycyclic

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

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