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## G = C3×C7⋊C18order 378 = 2·33·7

### Direct product of C3 and C7⋊C18

Aliases: C3×C7⋊C18, C212C18, C32.3F7, C7⋊C98C6, (C3×D7)⋊C9, C72(C3×C18), D72(C3×C9), C3.2(C3×F7), (C3×C21).3C6, C21.4(C3×C6), (C32×D7).2C3, (C3×D7).4C32, (C3×C7⋊C9)⋊2C2, SmallGroup(378,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C3×C7⋊C18
 Chief series C1 — C7 — C21 — C3×C21 — C3×C7⋊C9 — C3×C7⋊C18
 Lower central C7 — C3×C7⋊C18
 Upper central C1 — C32

Generators and relations for C3×C7⋊C18
G = < a,b,c | a3=b7=c18=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C3×C7⋊C18
On 189 points
Generators in S189
(1 14 22)(2 15 23)(3 16 24)(4 17 25)(5 18 26)(6 10 27)(7 11 19)(8 12 20)(9 13 21)(28 176 166)(29 177 167)(30 178 168)(31 179 169)(32 180 170)(33 181 171)(34 182 154)(35 183 155)(36 184 156)(37 185 157)(38 186 158)(39 187 159)(40 188 160)(41 189 161)(42 172 162)(43 173 163)(44 174 164)(45 175 165)(46 130 68)(47 131 69)(48 132 70)(49 133 71)(50 134 72)(51 135 73)(52 118 74)(53 119 75)(54 120 76)(55 121 77)(56 122 78)(57 123 79)(58 124 80)(59 125 81)(60 126 64)(61 127 65)(62 128 66)(63 129 67)(82 150 106)(83 151 107)(84 152 108)(85 153 109)(86 136 110)(87 137 111)(88 138 112)(89 139 113)(90 140 114)(91 141 115)(92 142 116)(93 143 117)(94 144 100)(95 145 101)(96 146 102)(97 147 103)(98 148 104)(99 149 105)
(1 60 159 83 92 168 51)(2 84 52 160 169 61 93)(3 161 94 53 62 85 170)(4 54 171 95 86 162 63)(5 96 46 154 163 55 87)(6 155 88 47 56 97 164)(7 48 165 89 98 156 57)(8 90 58 166 157 49 99)(9 167 82 59 50 91 158)(10 35 138 131 122 147 44)(11 132 45 139 148 36 123)(12 140 124 28 37 133 149)(13 29 150 125 134 141 38)(14 126 39 151 142 30 135)(15 152 118 40 31 127 143)(16 41 144 119 128 153 32)(17 120 33 145 136 42 129)(18 146 130 34 43 121 137)(19 70 175 113 104 184 79)(20 114 80 176 185 71 105)(21 177 106 81 72 115 186)(22 64 187 107 116 178 73)(23 108 74 188 179 65 117)(24 189 100 75 66 109 180)(25 76 181 101 110 172 67)(26 102 68 182 173 77 111)(27 183 112 69 78 103 174)
(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)(172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189)

G:=sub<Sym(189)| (1,14,22)(2,15,23)(3,16,24)(4,17,25)(5,18,26)(6,10,27)(7,11,19)(8,12,20)(9,13,21)(28,176,166)(29,177,167)(30,178,168)(31,179,169)(32,180,170)(33,181,171)(34,182,154)(35,183,155)(36,184,156)(37,185,157)(38,186,158)(39,187,159)(40,188,160)(41,189,161)(42,172,162)(43,173,163)(44,174,164)(45,175,165)(46,130,68)(47,131,69)(48,132,70)(49,133,71)(50,134,72)(51,135,73)(52,118,74)(53,119,75)(54,120,76)(55,121,77)(56,122,78)(57,123,79)(58,124,80)(59,125,81)(60,126,64)(61,127,65)(62,128,66)(63,129,67)(82,150,106)(83,151,107)(84,152,108)(85,153,109)(86,136,110)(87,137,111)(88,138,112)(89,139,113)(90,140,114)(91,141,115)(92,142,116)(93,143,117)(94,144,100)(95,145,101)(96,146,102)(97,147,103)(98,148,104)(99,149,105), (1,60,159,83,92,168,51)(2,84,52,160,169,61,93)(3,161,94,53,62,85,170)(4,54,171,95,86,162,63)(5,96,46,154,163,55,87)(6,155,88,47,56,97,164)(7,48,165,89,98,156,57)(8,90,58,166,157,49,99)(9,167,82,59,50,91,158)(10,35,138,131,122,147,44)(11,132,45,139,148,36,123)(12,140,124,28,37,133,149)(13,29,150,125,134,141,38)(14,126,39,151,142,30,135)(15,152,118,40,31,127,143)(16,41,144,119,128,153,32)(17,120,33,145,136,42,129)(18,146,130,34,43,121,137)(19,70,175,113,104,184,79)(20,114,80,176,185,71,105)(21,177,106,81,72,115,186)(22,64,187,107,116,178,73)(23,108,74,188,179,65,117)(24,189,100,75,66,109,180)(25,76,181,101,110,172,67)(26,102,68,182,173,77,111)(27,183,112,69,78,103,174), (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)(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189)>;

G:=Group( (1,14,22)(2,15,23)(3,16,24)(4,17,25)(5,18,26)(6,10,27)(7,11,19)(8,12,20)(9,13,21)(28,176,166)(29,177,167)(30,178,168)(31,179,169)(32,180,170)(33,181,171)(34,182,154)(35,183,155)(36,184,156)(37,185,157)(38,186,158)(39,187,159)(40,188,160)(41,189,161)(42,172,162)(43,173,163)(44,174,164)(45,175,165)(46,130,68)(47,131,69)(48,132,70)(49,133,71)(50,134,72)(51,135,73)(52,118,74)(53,119,75)(54,120,76)(55,121,77)(56,122,78)(57,123,79)(58,124,80)(59,125,81)(60,126,64)(61,127,65)(62,128,66)(63,129,67)(82,150,106)(83,151,107)(84,152,108)(85,153,109)(86,136,110)(87,137,111)(88,138,112)(89,139,113)(90,140,114)(91,141,115)(92,142,116)(93,143,117)(94,144,100)(95,145,101)(96,146,102)(97,147,103)(98,148,104)(99,149,105), (1,60,159,83,92,168,51)(2,84,52,160,169,61,93)(3,161,94,53,62,85,170)(4,54,171,95,86,162,63)(5,96,46,154,163,55,87)(6,155,88,47,56,97,164)(7,48,165,89,98,156,57)(8,90,58,166,157,49,99)(9,167,82,59,50,91,158)(10,35,138,131,122,147,44)(11,132,45,139,148,36,123)(12,140,124,28,37,133,149)(13,29,150,125,134,141,38)(14,126,39,151,142,30,135)(15,152,118,40,31,127,143)(16,41,144,119,128,153,32)(17,120,33,145,136,42,129)(18,146,130,34,43,121,137)(19,70,175,113,104,184,79)(20,114,80,176,185,71,105)(21,177,106,81,72,115,186)(22,64,187,107,116,178,73)(23,108,74,188,179,65,117)(24,189,100,75,66,109,180)(25,76,181,101,110,172,67)(26,102,68,182,173,77,111)(27,183,112,69,78,103,174), (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)(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189) );

G=PermutationGroup([[(1,14,22),(2,15,23),(3,16,24),(4,17,25),(5,18,26),(6,10,27),(7,11,19),(8,12,20),(9,13,21),(28,176,166),(29,177,167),(30,178,168),(31,179,169),(32,180,170),(33,181,171),(34,182,154),(35,183,155),(36,184,156),(37,185,157),(38,186,158),(39,187,159),(40,188,160),(41,189,161),(42,172,162),(43,173,163),(44,174,164),(45,175,165),(46,130,68),(47,131,69),(48,132,70),(49,133,71),(50,134,72),(51,135,73),(52,118,74),(53,119,75),(54,120,76),(55,121,77),(56,122,78),(57,123,79),(58,124,80),(59,125,81),(60,126,64),(61,127,65),(62,128,66),(63,129,67),(82,150,106),(83,151,107),(84,152,108),(85,153,109),(86,136,110),(87,137,111),(88,138,112),(89,139,113),(90,140,114),(91,141,115),(92,142,116),(93,143,117),(94,144,100),(95,145,101),(96,146,102),(97,147,103),(98,148,104),(99,149,105)], [(1,60,159,83,92,168,51),(2,84,52,160,169,61,93),(3,161,94,53,62,85,170),(4,54,171,95,86,162,63),(5,96,46,154,163,55,87),(6,155,88,47,56,97,164),(7,48,165,89,98,156,57),(8,90,58,166,157,49,99),(9,167,82,59,50,91,158),(10,35,138,131,122,147,44),(11,132,45,139,148,36,123),(12,140,124,28,37,133,149),(13,29,150,125,134,141,38),(14,126,39,151,142,30,135),(15,152,118,40,31,127,143),(16,41,144,119,128,153,32),(17,120,33,145,136,42,129),(18,146,130,34,43,121,137),(19,70,175,113,104,184,79),(20,114,80,176,185,71,105),(21,177,106,81,72,115,186),(22,64,187,107,116,178,73),(23,108,74,188,179,65,117),(24,189,100,75,66,109,180),(25,76,181,101,110,172,67),(26,102,68,182,173,77,111),(27,183,112,69,78,103,174)], [(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),(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189)]])

63 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 7 9A ··· 9R 18A ··· 18R 21A ··· 21H order 1 2 3 ··· 3 6 ··· 6 7 9 ··· 9 18 ··· 18 21 ··· 21 size 1 7 1 ··· 1 7 ··· 7 6 7 ··· 7 7 ··· 7 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 6 6 6 type + + + image C1 C2 C3 C3 C6 C6 C9 C18 F7 C7⋊C18 C3×F7 kernel C3×C7⋊C18 C3×C7⋊C9 C7⋊C18 C32×D7 C7⋊C9 C3×C21 C3×D7 C21 C32 C3 C3 # reps 1 1 6 2 6 2 18 18 1 6 2

Matrix representation of C3×C7⋊C18 in GL7(𝔽127)

 19 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 126 0 1 0 0 0 0 126 0 0 1 0 0 0 126 0 0 0 1 0 0 126 0 0 0 0 1 0 126 0 0 0 0 0 1 126
,
 20 0 0 0 0 0 0 0 108 19 114 109 0 18 0 108 37 0 90 19 5 0 0 24 109 90 37 18 0 18 37 90 109 24 0 0 5 19 90 0 37 108 0 18 0 109 114 19 108

G:=sub<GL(7,GF(127))| [19,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,126,126,126,126,126,126],[20,0,0,0,0,0,0,0,108,108,0,18,5,18,0,19,37,24,37,19,0,0,114,0,109,90,90,109,0,109,90,90,109,0,114,0,0,19,37,24,37,19,0,18,5,18,0,108,108] >;

C3×C7⋊C18 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes C_{18}
% in TeX

G:=Group("C3xC7:C18");
// GroupNames label

G:=SmallGroup(378,10);
// by ID

G=gap.SmallGroup(378,10);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,96,8104,2709]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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