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## G = C42×D11order 352 = 25·11

### Direct product of C42 and D11

Series: Derived Chief Lower central Upper central

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

Generators and relations for C42×D11
G = < a,b,c,d | a4=b4=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 498 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C11, C42, C42, C22×C4, D11, C22, C2×C42, Dic11, C44, D22, C2×C22, C4×D11, C2×Dic11, C2×C44, C22×D11, C4×Dic11, C4×C44, C2×C4×D11, C42×D11
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, D11, C2×C42, D22, C4×D11, C22×D11, C2×C4×D11, C42×D11

Smallest permutation representation of C42×D11
On 176 points
Generators in S176
(1 109 21 98)(2 110 22 99)(3 100 12 89)(4 101 13 90)(5 102 14 91)(6 103 15 92)(7 104 16 93)(8 105 17 94)(9 106 18 95)(10 107 19 96)(11 108 20 97)(23 122 34 111)(24 123 35 112)(25 124 36 113)(26 125 37 114)(27 126 38 115)(28 127 39 116)(29 128 40 117)(30 129 41 118)(31 130 42 119)(32 131 43 120)(33 132 44 121)(45 144 56 133)(46 145 57 134)(47 146 58 135)(48 147 59 136)(49 148 60 137)(50 149 61 138)(51 150 62 139)(52 151 63 140)(53 152 64 141)(54 153 65 142)(55 154 66 143)(67 166 78 155)(68 167 79 156)(69 168 80 157)(70 169 81 158)(71 170 82 159)(72 171 83 160)(73 172 84 161)(74 173 85 162)(75 174 86 163)(76 175 87 164)(77 176 88 165)
(1 87 32 65)(2 88 33 66)(3 78 23 56)(4 79 24 57)(5 80 25 58)(6 81 26 59)(7 82 27 60)(8 83 28 61)(9 84 29 62)(10 85 30 63)(11 86 31 64)(12 67 34 45)(13 68 35 46)(14 69 36 47)(15 70 37 48)(16 71 38 49)(17 72 39 50)(18 73 40 51)(19 74 41 52)(20 75 42 53)(21 76 43 54)(22 77 44 55)(89 166 111 144)(90 167 112 145)(91 168 113 146)(92 169 114 147)(93 170 115 148)(94 171 116 149)(95 172 117 150)(96 173 118 151)(97 174 119 152)(98 175 120 153)(99 176 121 154)(100 155 122 133)(101 156 123 134)(102 157 124 135)(103 158 125 136)(104 159 126 137)(105 160 127 138)(106 161 128 139)(107 162 129 140)(108 163 130 141)(109 164 131 142)(110 165 132 143)
(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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 33)(20 32)(21 31)(22 30)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 88)(53 87)(54 86)(55 85)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 77)(64 76)(65 75)(66 74)(89 128)(90 127)(91 126)(92 125)(93 124)(94 123)(95 122)(96 132)(97 131)(98 130)(99 129)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 121)(108 120)(109 119)(110 118)(133 172)(134 171)(135 170)(136 169)(137 168)(138 167)(139 166)(140 176)(141 175)(142 174)(143 173)(144 161)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 165)(152 164)(153 163)(154 162)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4X 11A ··· 11E 22A ··· 22O 44A ··· 44BH order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 11 11 11 11 1 ··· 1 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D11 D22 C4×D11 kernel C42×D11 C4×Dic11 C4×C44 C2×C4×D11 C4×D11 C42 C2×C4 C4 # reps 1 3 1 3 24 5 15 60

Matrix representation of C42×D11 in GL3(𝔽89) generated by

 34 0 0 0 1 0 0 0 1
,
 1 0 0 0 55 0 0 0 55
,
 1 0 0 0 55 1 0 88 0
,
 1 0 0 0 0 88 0 88 0
G:=sub<GL(3,GF(89))| [34,0,0,0,1,0,0,0,1],[1,0,0,0,55,0,0,0,55],[1,0,0,0,55,88,0,1,0],[1,0,0,0,0,88,0,88,0] >;

C42×D11 in GAP, Magma, Sage, TeX

C_4^2\times D_{11}
% in TeX

G:=Group("C4^2xD11");
// GroupNames label

G:=SmallGroup(352,66);
// by ID

G=gap.SmallGroup(352,66);
# by ID

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

G:=Group<a,b,c,d|a^4=b^4=c^11=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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