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## G = C2×C44.C4order 352 = 25·11

### Direct product of C2 and C44.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C44.C4
 Chief series C1 — C11 — C22 — C44 — C11⋊C8 — C2×C11⋊C8 — C2×C44.C4
 Lower central C11 — C22 — C2×C44.C4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C44.C4
G = < a,b,c | a2=b44=1, c4=b22, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 186 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C11, C2×C8, M4(2), C22×C4, C22, C22, C22, C2×M4(2), C44, C44, C2×C22, C2×C22, C2×C22, C11⋊C8, C2×C44, C2×C44, C22×C22, C2×C11⋊C8, C44.C4, C22×C44, C2×C44.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, D11, C2×M4(2), Dic11, D22, C2×Dic11, C22×D11, C44.C4, C22×Dic11, C2×C44.C4

Smallest permutation representation of C2×C44.C4
On 176 points
Generators in S176
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(89 157)(90 158)(91 159)(92 160)(93 161)(94 162)(95 163)(96 164)(97 165)(98 166)(99 167)(100 168)(101 169)(102 170)(103 171)(104 172)(105 173)(106 174)(107 175)(108 176)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)(121 145)(122 146)(123 147)(124 148)(125 149)(126 150)(127 151)(128 152)(129 153)(130 154)(131 155)(132 156)
(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)
(1 106 34 117 23 128 12 95)(2 105 35 116 24 127 13 94)(3 104 36 115 25 126 14 93)(4 103 37 114 26 125 15 92)(5 102 38 113 27 124 16 91)(6 101 39 112 28 123 17 90)(7 100 40 111 29 122 18 89)(8 99 41 110 30 121 19 132)(9 98 42 109 31 120 20 131)(10 97 43 108 32 119 21 130)(11 96 44 107 33 118 22 129)(45 165 78 176 67 143 56 154)(46 164 79 175 68 142 57 153)(47 163 80 174 69 141 58 152)(48 162 81 173 70 140 59 151)(49 161 82 172 71 139 60 150)(50 160 83 171 72 138 61 149)(51 159 84 170 73 137 62 148)(52 158 85 169 74 136 63 147)(53 157 86 168 75 135 64 146)(54 156 87 167 76 134 65 145)(55 155 88 166 77 133 66 144)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A ··· 8H 11A ··· 11E 22A ··· 22AI 44A ··· 44AN order 1 2 2 2 2 2 4 4 4 4 4 4 8 ··· 8 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 2 2 1 1 1 1 2 2 22 ··· 22 2 ··· 2 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 M4(2) D11 Dic11 D22 Dic11 C44.C4 kernel C2×C44.C4 C2×C11⋊C8 C44.C4 C22×C44 C2×C44 C22×C22 C22 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 4 5 15 15 5 40

Matrix representation of C2×C44.C4 in GL3(𝔽89) generated by

 88 0 0 0 88 0 0 0 88
,
 1 0 0 0 9 4 0 0 10
,
 55 0 0 0 86 87 0 66 3
G:=sub<GL(3,GF(89))| [88,0,0,0,88,0,0,0,88],[1,0,0,0,9,0,0,4,10],[55,0,0,0,86,66,0,87,3] >;

C2×C44.C4 in GAP, Magma, Sage, TeX

C_2\times C_{44}.C_4
% in TeX

G:=Group("C2xC44.C4");
// GroupNames label

G:=SmallGroup(352,116);
// by ID

G=gap.SmallGroup(352,116);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,362,69,11525]);
// Polycyclic

G:=Group<a,b,c|a^2=b^44=1,c^4=b^22,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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