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## G = C2×D92order 368 = 24·23

### Direct product of C2 and D92

Aliases: C2×D92, C42D46, C461D4, C922C22, D461C22, C46.3C23, C22.10D46, C231(C2×D4), (C2×C92)⋊3C2, (C2×C4)⋊2D23, (C22×D23)⋊1C2, C2.4(C22×D23), (C2×C46).10C22, SmallGroup(368,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C46 — C2×D92
 Chief series C1 — C23 — C46 — D46 — C22×D23 — C2×D92
 Lower central C23 — C46 — C2×D92
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D92
G = < a,b,c | a2=b92=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 664 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C2×D4, C23, D23, C46, C46, C92, D46, D46, C2×C46, D92, C2×C92, C22×D23, C2×D92
Quotients: C1, C2, C22, D4, C23, C2×D4, D23, D46, D92, C22×D23, C2×D92

Smallest permutation representation of C2×D92
On 184 points
Generators in S184
(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)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 181)(74 182)(75 183)(76 184)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)
(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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(125 184)(126 183)(127 182)(128 181)(129 180)(130 179)(131 178)(132 177)(133 176)(134 175)(135 174)(136 173)(137 172)(138 171)(139 170)(140 169)(141 168)(142 167)(143 166)(144 165)(145 164)(146 163)(147 162)(148 161)(149 160)(150 159)(151 158)(152 157)(153 156)(154 155)

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

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

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

98 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 23A ··· 23K 46A ··· 46AG 92A ··· 92AR order 1 2 2 2 2 2 2 2 4 4 23 ··· 23 46 ··· 46 92 ··· 92 size 1 1 1 1 46 46 46 46 2 2 2 ··· 2 2 ··· 2 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D23 D46 D46 D92 kernel C2×D92 D92 C2×C92 C22×D23 C46 C2×C4 C4 C22 C2 # reps 1 4 1 2 2 11 22 11 44

Matrix representation of C2×D92 in GL3(𝔽277) generated by

 276 0 0 0 276 0 0 0 276
,
 276 0 0 0 251 98 0 13 196
,
 1 0 0 0 48 222 0 198 229
G:=sub<GL(3,GF(277))| [276,0,0,0,276,0,0,0,276],[276,0,0,0,251,13,0,98,196],[1,0,0,0,48,198,0,222,229] >;

C2×D92 in GAP, Magma, Sage, TeX

C_2\times D_{92}
% in TeX

G:=Group("C2xD92");
// GroupNames label

G:=SmallGroup(368,29);
// by ID

G=gap.SmallGroup(368,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-23,182,42,8804]);
// Polycyclic

G:=Group<a,b,c|a^2=b^92=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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