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G = C4×D47order 376 = 23·47

Direct product of C4 and D47

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D47, D94.C2, C1882C2, C2.1D94, Dic472C2, C94.2C22, C471(C2×C4), SmallGroup(376,4)

Series: Derived Chief Lower central Upper central

C1C47 — C4×D47
C1C47C94D94 — C4×D47
C47 — C4×D47
C1C4

Generators and relations for C4×D47
 G = < a,b,c | a4=b47=c2=1, ab=ba, ac=ca, cbc=b-1 >

47C2
47C2
47C22
47C4
47C2×C4

Smallest permutation representation of C4×D47
On 188 points
Generators in S188
(1 187 60 115)(2 188 61 116)(3 142 62 117)(4 143 63 118)(5 144 64 119)(6 145 65 120)(7 146 66 121)(8 147 67 122)(9 148 68 123)(10 149 69 124)(11 150 70 125)(12 151 71 126)(13 152 72 127)(14 153 73 128)(15 154 74 129)(16 155 75 130)(17 156 76 131)(18 157 77 132)(19 158 78 133)(20 159 79 134)(21 160 80 135)(22 161 81 136)(23 162 82 137)(24 163 83 138)(25 164 84 139)(26 165 85 140)(27 166 86 141)(28 167 87 95)(29 168 88 96)(30 169 89 97)(31 170 90 98)(32 171 91 99)(33 172 92 100)(34 173 93 101)(35 174 94 102)(36 175 48 103)(37 176 49 104)(38 177 50 105)(39 178 51 106)(40 179 52 107)(41 180 53 108)(42 181 54 109)(43 182 55 110)(44 183 56 111)(45 184 57 112)(46 185 58 113)(47 186 59 114)
(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)(142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(48 71)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(72 94)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(135 141)(136 140)(137 139)(142 184)(143 183)(144 182)(145 181)(146 180)(147 179)(148 178)(149 177)(150 176)(151 175)(152 174)(153 173)(154 172)(155 171)(156 170)(157 169)(158 168)(159 167)(160 166)(161 165)(162 164)(185 188)(186 187)

G:=sub<Sym(188)| (1,187,60,115)(2,188,61,116)(3,142,62,117)(4,143,63,118)(5,144,64,119)(6,145,65,120)(7,146,66,121)(8,147,67,122)(9,148,68,123)(10,149,69,124)(11,150,70,125)(12,151,71,126)(13,152,72,127)(14,153,73,128)(15,154,74,129)(16,155,75,130)(17,156,76,131)(18,157,77,132)(19,158,78,133)(20,159,79,134)(21,160,80,135)(22,161,81,136)(23,162,82,137)(24,163,83,138)(25,164,84,139)(26,165,85,140)(27,166,86,141)(28,167,87,95)(29,168,88,96)(30,169,89,97)(31,170,90,98)(32,171,91,99)(33,172,92,100)(34,173,93,101)(35,174,94,102)(36,175,48,103)(37,176,49,104)(38,177,50,105)(39,178,51,106)(40,179,52,107)(41,180,53,108)(42,181,54,109)(43,182,55,110)(44,183,56,111)(45,184,57,112)(46,185,58,113)(47,186,59,114), (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)(142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(135,141)(136,140)(137,139)(142,184)(143,183)(144,182)(145,181)(146,180)(147,179)(148,178)(149,177)(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(185,188)(186,187)>;

G:=Group( (1,187,60,115)(2,188,61,116)(3,142,62,117)(4,143,63,118)(5,144,64,119)(6,145,65,120)(7,146,66,121)(8,147,67,122)(9,148,68,123)(10,149,69,124)(11,150,70,125)(12,151,71,126)(13,152,72,127)(14,153,73,128)(15,154,74,129)(16,155,75,130)(17,156,76,131)(18,157,77,132)(19,158,78,133)(20,159,79,134)(21,160,80,135)(22,161,81,136)(23,162,82,137)(24,163,83,138)(25,164,84,139)(26,165,85,140)(27,166,86,141)(28,167,87,95)(29,168,88,96)(30,169,89,97)(31,170,90,98)(32,171,91,99)(33,172,92,100)(34,173,93,101)(35,174,94,102)(36,175,48,103)(37,176,49,104)(38,177,50,105)(39,178,51,106)(40,179,52,107)(41,180,53,108)(42,181,54,109)(43,182,55,110)(44,183,56,111)(45,184,57,112)(46,185,58,113)(47,186,59,114), (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)(142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)(135,141)(136,140)(137,139)(142,184)(143,183)(144,182)(145,181)(146,180)(147,179)(148,178)(149,177)(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(185,188)(186,187) );

G=PermutationGroup([(1,187,60,115),(2,188,61,116),(3,142,62,117),(4,143,63,118),(5,144,64,119),(6,145,65,120),(7,146,66,121),(8,147,67,122),(9,148,68,123),(10,149,69,124),(11,150,70,125),(12,151,71,126),(13,152,72,127),(14,153,73,128),(15,154,74,129),(16,155,75,130),(17,156,76,131),(18,157,77,132),(19,158,78,133),(20,159,79,134),(21,160,80,135),(22,161,81,136),(23,162,82,137),(24,163,83,138),(25,164,84,139),(26,165,85,140),(27,166,86,141),(28,167,87,95),(29,168,88,96),(30,169,89,97),(31,170,90,98),(32,171,91,99),(33,172,92,100),(34,173,93,101),(35,174,94,102),(36,175,48,103),(37,176,49,104),(38,177,50,105),(39,178,51,106),(40,179,52,107),(41,180,53,108),(42,181,54,109),(43,182,55,110),(44,183,56,111),(45,184,57,112),(46,185,58,113),(47,186,59,114)], [(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),(142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(48,71),(49,70),(50,69),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,60),(72,94),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(95,134),(96,133),(97,132),(98,131),(99,130),(100,129),(101,128),(102,127),(103,126),(104,125),(105,124),(106,123),(107,122),(108,121),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115),(135,141),(136,140),(137,139),(142,184),(143,183),(144,182),(145,181),(146,180),(147,179),(148,178),(149,177),(150,176),(151,175),(152,174),(153,173),(154,172),(155,171),(156,170),(157,169),(158,168),(159,167),(160,166),(161,165),(162,164),(185,188),(186,187)])

100 conjugacy classes

class 1 2A2B2C4A4B4C4D47A···47W94A···94W188A···188AT
order1222444447···4794···94188···188
size1147471147472···22···22···2

100 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D47D94C4×D47
kernelC4×D47Dic47C188D94D47C4C2C1
# reps11114232346

Matrix representation of C4×D47 in GL3(𝔽941) generated by

84400
010
001
,
100
001
0940334
,
94000
001
010
G:=sub<GL(3,GF(941))| [844,0,0,0,1,0,0,0,1],[1,0,0,0,0,940,0,1,334],[940,0,0,0,0,1,0,1,0] >;

C4×D47 in GAP, Magma, Sage, TeX

C_4\times D_{47}
% in TeX

G:=Group("C4xD47");
// GroupNames label

G:=SmallGroup(376,4);
// by ID

G=gap.SmallGroup(376,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-47,21,5891]);
// Polycyclic

G:=Group<a,b,c|a^4=b^47=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D47 in TeX

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