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## G = D4×C46order 368 = 24·23

### Direct product of C46 and D4

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

Aliases: D4×C46, C23⋊C46, C924C22, C46.11C23, C4⋊(C2×C46), (C2×C92)⋊6C2, (C2×C4)⋊2C46, C22⋊(C2×C46), (C2×C46)⋊2C22, (C22×C46)⋊1C2, C2.1(C22×C46), SmallGroup(368,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C46
 Chief series C1 — C2 — C46 — C2×C46 — D4×C23 — D4×C46
 Lower central C1 — C2 — D4×C46
 Upper central C1 — C2×C46 — D4×C46

Generators and relations for D4×C46
G = < a,b,c | a46=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C2×D4, C23, C46, C46, C46, C92, C2×C46, C2×C46, C2×C46, C2×C92, D4×C23, C22×C46, D4×C46
Quotients: C1, C2, C22, D4, C23, C2×D4, C23, C46, C2×C46, D4×C23, C22×C46, D4×C46

Smallest permutation representation of D4×C46
On 184 points
Generators in S184
(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)
(1 132 83 150)(2 133 84 151)(3 134 85 152)(4 135 86 153)(5 136 87 154)(6 137 88 155)(7 138 89 156)(8 93 90 157)(9 94 91 158)(10 95 92 159)(11 96 47 160)(12 97 48 161)(13 98 49 162)(14 99 50 163)(15 100 51 164)(16 101 52 165)(17 102 53 166)(18 103 54 167)(19 104 55 168)(20 105 56 169)(21 106 57 170)(22 107 58 171)(23 108 59 172)(24 109 60 173)(25 110 61 174)(26 111 62 175)(27 112 63 176)(28 113 64 177)(29 114 65 178)(30 115 66 179)(31 116 67 180)(32 117 68 181)(33 118 69 182)(34 119 70 183)(35 120 71 184)(36 121 72 139)(37 122 73 140)(38 123 74 141)(39 124 75 142)(40 125 76 143)(41 126 77 144)(42 127 78 145)(43 128 79 146)(44 129 80 147)(45 130 81 148)(46 131 82 149)
(1 109)(2 110)(3 111)(4 112)(5 113)(6 114)(7 115)(8 116)(9 117)(10 118)(11 119)(12 120)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 183)(48 184)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 163)(74 164)(75 165)(76 166)(77 167)(78 168)(79 169)(80 170)(81 171)(82 172)(83 173)(84 174)(85 175)(86 176)(87 177)(88 178)(89 179)(90 180)(91 181)(92 182)

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

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

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

230 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 23A ··· 23V 46A ··· 46BN 46BO ··· 46EX 92A ··· 92AR order 1 2 2 2 2 2 2 2 4 4 23 ··· 23 46 ··· 46 46 ··· 46 92 ··· 92 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

230 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C23 C46 C46 C46 D4 D4×C23 kernel D4×C46 C2×C92 D4×C23 C22×C46 C2×D4 C2×C4 D4 C23 C46 C2 # reps 1 1 4 2 22 22 88 44 2 44

Matrix representation of D4×C46 in GL3(𝔽277) generated by

 276 0 0 0 30 0 0 0 30
,
 1 0 0 0 138 34 0 2 139
,
 1 0 0 0 138 35 0 2 139
G:=sub<GL(3,GF(277))| [276,0,0,0,30,0,0,0,30],[1,0,0,0,138,2,0,34,139],[1,0,0,0,138,2,0,35,139] >;

D4×C46 in GAP, Magma, Sage, TeX

D_4\times C_{46}
% in TeX

G:=Group("D4xC46");
// GroupNames label

G:=SmallGroup(368,38);
// by ID

G=gap.SmallGroup(368,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-23,-2,1861]);
// Polycyclic

G:=Group<a,b,c|a^46=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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