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## G = C3×C7⋊C9order 189 = 33·7

### Direct product of C3 and C7⋊C9

Aliases: C3×C7⋊C9, C21⋊C9, C21.4C32, C72(C3×C9), (C3×C21).2C3, C32.2(C7⋊C3), C3.2(C3×C7⋊C3), SmallGroup(189,6)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C7⋊C9
G = < a,b,c | a3=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C3×C7⋊C9
Regular action on 189 points
Generators in S189
(1 178 46)(2 179 47)(3 180 48)(4 172 49)(5 173 50)(6 174 51)(7 175 52)(8 176 53)(9 177 54)(10 119 78)(11 120 79)(12 121 80)(13 122 81)(14 123 73)(15 124 74)(16 125 75)(17 126 76)(18 118 77)(19 185 37)(20 186 38)(21 187 39)(22 188 40)(23 189 41)(24 181 42)(25 182 43)(26 183 44)(27 184 45)(28 149 101)(29 150 102)(30 151 103)(31 152 104)(32 153 105)(33 145 106)(34 146 107)(35 147 108)(36 148 100)(55 86 132)(56 87 133)(57 88 134)(58 89 135)(59 90 127)(60 82 128)(61 83 129)(62 84 130)(63 85 131)(64 166 115)(65 167 116)(66 168 117)(67 169 109)(68 170 110)(69 171 111)(70 163 112)(71 164 113)(72 165 114)(91 137 159)(92 138 160)(93 139 161)(94 140 162)(95 141 154)(96 142 155)(97 143 156)(98 144 157)(99 136 158)
(1 34 17 127 95 72 182)(2 96 35 64 18 183 128)(3 10 97 184 36 129 65)(4 28 11 130 98 66 185)(5 99 29 67 12 186 131)(6 13 91 187 30 132 68)(7 31 14 133 92 69 188)(8 93 32 70 15 189 134)(9 16 94 181 33 135 71)(19 49 101 79 84 157 117)(20 85 50 158 102 109 80)(21 103 86 110 51 81 159)(22 52 104 73 87 160 111)(23 88 53 161 105 112 74)(24 106 89 113 54 75 162)(25 46 107 76 90 154 114)(26 82 47 155 108 115 77)(27 100 83 116 48 78 156)(37 172 149 120 62 144 168)(38 63 173 136 150 169 121)(39 151 55 170 174 122 137)(40 175 152 123 56 138 171)(41 57 176 139 153 163 124)(42 145 58 164 177 125 140)(43 178 146 126 59 141 165)(44 60 179 142 147 166 118)(45 148 61 167 180 119 143)
(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)(172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189)

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

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

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

C3×C7⋊C9 is a maximal subgroup of   D21⋊C9

45 conjugacy classes

 class 1 3A ··· 3H 7A 7B 9A ··· 9R 21A ··· 21P order 1 3 ··· 3 7 7 9 ··· 9 21 ··· 21 size 1 1 ··· 1 3 3 7 ··· 7 3 ··· 3

45 irreducible representations

 dim 1 1 1 1 3 3 3 type + image C1 C3 C3 C9 C7⋊C3 C7⋊C9 C3×C7⋊C3 kernel C3×C7⋊C9 C7⋊C9 C3×C21 C21 C32 C3 C3 # reps 1 6 2 18 2 12 4

Matrix representation of C3×C7⋊C9 in GL4(𝔽127) generated by

 107 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 126 104 1 0 0 104 1 0 126 105 1
,
 107 0 0 0 0 67 17 94 0 3 43 124 0 43 77 17
G:=sub<GL(4,GF(127))| [107,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,126,0,126,0,104,104,105,0,1,1,1],[107,0,0,0,0,67,3,43,0,17,43,77,0,94,124,17] >;

C3×C7⋊C9 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes C_9
% in TeX

G:=Group("C3xC7:C9");
// GroupNames label

G:=SmallGroup(189,6);
// by ID

G=gap.SmallGroup(189,6);
# by ID

G:=PCGroup([4,-3,-3,-3,-7,36,867]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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