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## G = C4×C7⋊C9order 252 = 22·32·7

### Direct product of C4 and C7⋊C9

Aliases: C4×C7⋊C9, C28⋊C9, C72C36, C84.C3, C42.6C6, C14.2C18, C21.2C12, C12.(C7⋊C3), C2.(C2×C7⋊C9), C3.(C4×C7⋊C3), C6.2(C2×C7⋊C3), (C2×C7⋊C9).2C2, SmallGroup(252,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C4×C7⋊C9
 Chief series C1 — C7 — C21 — C42 — C2×C7⋊C9 — C4×C7⋊C9
 Lower central C7 — C4×C7⋊C9
 Upper central C1 — C12

Generators and relations for C4×C7⋊C9
G = < a,b,c | a4=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C4×C7⋊C9
Regular action on 252 points
Generators in S252
(1 165 110 42)(2 166 111 43)(3 167 112 44)(4 168 113 45)(5 169 114 37)(6 170 115 38)(7 171 116 39)(8 163 117 40)(9 164 109 41)(10 180 240 19)(11 172 241 20)(12 173 242 21)(13 174 243 22)(14 175 235 23)(15 176 236 24)(16 177 237 25)(17 178 238 26)(18 179 239 27)(28 93 228 212)(29 94 229 213)(30 95 230 214)(31 96 231 215)(32 97 232 216)(33 98 233 208)(34 99 234 209)(35 91 226 210)(36 92 227 211)(46 221 156 81)(47 222 157 73)(48 223 158 74)(49 224 159 75)(50 225 160 76)(51 217 161 77)(52 218 162 78)(53 219 154 79)(54 220 155 80)(55 181 200 124)(56 182 201 125)(57 183 202 126)(58 184 203 118)(59 185 204 119)(60 186 205 120)(61 187 206 121)(62 188 207 122)(63 189 199 123)(64 252 129 151)(65 244 130 152)(66 245 131 153)(67 246 132 145)(68 247 133 146)(69 248 134 147)(70 249 135 148)(71 250 127 149)(72 251 128 150)(82 138 192 104)(83 139 193 105)(84 140 194 106)(85 141 195 107)(86 142 196 108)(87 143 197 100)(88 144 198 101)(89 136 190 102)(90 137 191 103)
(1 68 119 105 81 178 33)(2 73 69 179 120 34 106)(3 121 74 35 70 107 180)(4 71 122 108 75 172 36)(5 76 72 173 123 28 100)(6 124 77 29 64 101 174)(7 65 125 102 78 175 30)(8 79 66 176 126 31 103)(9 118 80 32 67 104 177)(10 44 206 158 210 148 195)(11 211 45 149 207 196 159)(12 199 212 197 37 160 150)(13 38 200 161 213 151 198)(14 214 39 152 201 190 162)(15 202 215 191 40 154 153)(16 41 203 155 216 145 192)(17 208 42 146 204 193 156)(18 205 209 194 43 157 147)(19 112 187 223 226 135 141)(20 227 113 127 188 142 224)(21 189 228 143 114 225 128)(22 115 181 217 229 129 144)(23 230 116 130 182 136 218)(24 183 231 137 117 219 131)(25 109 184 220 232 132 138)(26 233 110 133 185 139 221)(27 186 234 140 111 222 134)(46 238 98 165 247 59 83)(47 248 239 60 99 84 166)(48 91 249 85 240 167 61)(49 241 92 168 250 62 86)(50 251 242 63 93 87 169)(51 94 252 88 243 170 55)(52 235 95 171 244 56 89)(53 245 236 57 96 90 163)(54 97 246 82 237 164 58)
(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)(190 191 192 193 194 195 196 197 198)(199 200 201 202 203 204 205 206 207)(208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225)(226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243)(244 245 246 247 248 249 250 251 252)

G:=sub<Sym(252)| (1,165,110,42)(2,166,111,43)(3,167,112,44)(4,168,113,45)(5,169,114,37)(6,170,115,38)(7,171,116,39)(8,163,117,40)(9,164,109,41)(10,180,240,19)(11,172,241,20)(12,173,242,21)(13,174,243,22)(14,175,235,23)(15,176,236,24)(16,177,237,25)(17,178,238,26)(18,179,239,27)(28,93,228,212)(29,94,229,213)(30,95,230,214)(31,96,231,215)(32,97,232,216)(33,98,233,208)(34,99,234,209)(35,91,226,210)(36,92,227,211)(46,221,156,81)(47,222,157,73)(48,223,158,74)(49,224,159,75)(50,225,160,76)(51,217,161,77)(52,218,162,78)(53,219,154,79)(54,220,155,80)(55,181,200,124)(56,182,201,125)(57,183,202,126)(58,184,203,118)(59,185,204,119)(60,186,205,120)(61,187,206,121)(62,188,207,122)(63,189,199,123)(64,252,129,151)(65,244,130,152)(66,245,131,153)(67,246,132,145)(68,247,133,146)(69,248,134,147)(70,249,135,148)(71,250,127,149)(72,251,128,150)(82,138,192,104)(83,139,193,105)(84,140,194,106)(85,141,195,107)(86,142,196,108)(87,143,197,100)(88,144,198,101)(89,136,190,102)(90,137,191,103), (1,68,119,105,81,178,33)(2,73,69,179,120,34,106)(3,121,74,35,70,107,180)(4,71,122,108,75,172,36)(5,76,72,173,123,28,100)(6,124,77,29,64,101,174)(7,65,125,102,78,175,30)(8,79,66,176,126,31,103)(9,118,80,32,67,104,177)(10,44,206,158,210,148,195)(11,211,45,149,207,196,159)(12,199,212,197,37,160,150)(13,38,200,161,213,151,198)(14,214,39,152,201,190,162)(15,202,215,191,40,154,153)(16,41,203,155,216,145,192)(17,208,42,146,204,193,156)(18,205,209,194,43,157,147)(19,112,187,223,226,135,141)(20,227,113,127,188,142,224)(21,189,228,143,114,225,128)(22,115,181,217,229,129,144)(23,230,116,130,182,136,218)(24,183,231,137,117,219,131)(25,109,184,220,232,132,138)(26,233,110,133,185,139,221)(27,186,234,140,111,222,134)(46,238,98,165,247,59,83)(47,248,239,60,99,84,166)(48,91,249,85,240,167,61)(49,241,92,168,250,62,86)(50,251,242,63,93,87,169)(51,94,252,88,243,170,55)(52,235,95,171,244,56,89)(53,245,236,57,96,90,163)(54,97,246,82,237,164,58), (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)(190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207)(208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243)(244,245,246,247,248,249,250,251,252)>;

G:=Group( (1,165,110,42)(2,166,111,43)(3,167,112,44)(4,168,113,45)(5,169,114,37)(6,170,115,38)(7,171,116,39)(8,163,117,40)(9,164,109,41)(10,180,240,19)(11,172,241,20)(12,173,242,21)(13,174,243,22)(14,175,235,23)(15,176,236,24)(16,177,237,25)(17,178,238,26)(18,179,239,27)(28,93,228,212)(29,94,229,213)(30,95,230,214)(31,96,231,215)(32,97,232,216)(33,98,233,208)(34,99,234,209)(35,91,226,210)(36,92,227,211)(46,221,156,81)(47,222,157,73)(48,223,158,74)(49,224,159,75)(50,225,160,76)(51,217,161,77)(52,218,162,78)(53,219,154,79)(54,220,155,80)(55,181,200,124)(56,182,201,125)(57,183,202,126)(58,184,203,118)(59,185,204,119)(60,186,205,120)(61,187,206,121)(62,188,207,122)(63,189,199,123)(64,252,129,151)(65,244,130,152)(66,245,131,153)(67,246,132,145)(68,247,133,146)(69,248,134,147)(70,249,135,148)(71,250,127,149)(72,251,128,150)(82,138,192,104)(83,139,193,105)(84,140,194,106)(85,141,195,107)(86,142,196,108)(87,143,197,100)(88,144,198,101)(89,136,190,102)(90,137,191,103), (1,68,119,105,81,178,33)(2,73,69,179,120,34,106)(3,121,74,35,70,107,180)(4,71,122,108,75,172,36)(5,76,72,173,123,28,100)(6,124,77,29,64,101,174)(7,65,125,102,78,175,30)(8,79,66,176,126,31,103)(9,118,80,32,67,104,177)(10,44,206,158,210,148,195)(11,211,45,149,207,196,159)(12,199,212,197,37,160,150)(13,38,200,161,213,151,198)(14,214,39,152,201,190,162)(15,202,215,191,40,154,153)(16,41,203,155,216,145,192)(17,208,42,146,204,193,156)(18,205,209,194,43,157,147)(19,112,187,223,226,135,141)(20,227,113,127,188,142,224)(21,189,228,143,114,225,128)(22,115,181,217,229,129,144)(23,230,116,130,182,136,218)(24,183,231,137,117,219,131)(25,109,184,220,232,132,138)(26,233,110,133,185,139,221)(27,186,234,140,111,222,134)(46,238,98,165,247,59,83)(47,248,239,60,99,84,166)(48,91,249,85,240,167,61)(49,241,92,168,250,62,86)(50,251,242,63,93,87,169)(51,94,252,88,243,170,55)(52,235,95,171,244,56,89)(53,245,236,57,96,90,163)(54,97,246,82,237,164,58), (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)(190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207)(208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243)(244,245,246,247,248,249,250,251,252) );

G=PermutationGroup([[(1,165,110,42),(2,166,111,43),(3,167,112,44),(4,168,113,45),(5,169,114,37),(6,170,115,38),(7,171,116,39),(8,163,117,40),(9,164,109,41),(10,180,240,19),(11,172,241,20),(12,173,242,21),(13,174,243,22),(14,175,235,23),(15,176,236,24),(16,177,237,25),(17,178,238,26),(18,179,239,27),(28,93,228,212),(29,94,229,213),(30,95,230,214),(31,96,231,215),(32,97,232,216),(33,98,233,208),(34,99,234,209),(35,91,226,210),(36,92,227,211),(46,221,156,81),(47,222,157,73),(48,223,158,74),(49,224,159,75),(50,225,160,76),(51,217,161,77),(52,218,162,78),(53,219,154,79),(54,220,155,80),(55,181,200,124),(56,182,201,125),(57,183,202,126),(58,184,203,118),(59,185,204,119),(60,186,205,120),(61,187,206,121),(62,188,207,122),(63,189,199,123),(64,252,129,151),(65,244,130,152),(66,245,131,153),(67,246,132,145),(68,247,133,146),(69,248,134,147),(70,249,135,148),(71,250,127,149),(72,251,128,150),(82,138,192,104),(83,139,193,105),(84,140,194,106),(85,141,195,107),(86,142,196,108),(87,143,197,100),(88,144,198,101),(89,136,190,102),(90,137,191,103)], [(1,68,119,105,81,178,33),(2,73,69,179,120,34,106),(3,121,74,35,70,107,180),(4,71,122,108,75,172,36),(5,76,72,173,123,28,100),(6,124,77,29,64,101,174),(7,65,125,102,78,175,30),(8,79,66,176,126,31,103),(9,118,80,32,67,104,177),(10,44,206,158,210,148,195),(11,211,45,149,207,196,159),(12,199,212,197,37,160,150),(13,38,200,161,213,151,198),(14,214,39,152,201,190,162),(15,202,215,191,40,154,153),(16,41,203,155,216,145,192),(17,208,42,146,204,193,156),(18,205,209,194,43,157,147),(19,112,187,223,226,135,141),(20,227,113,127,188,142,224),(21,189,228,143,114,225,128),(22,115,181,217,229,129,144),(23,230,116,130,182,136,218),(24,183,231,137,117,219,131),(25,109,184,220,232,132,138),(26,233,110,133,185,139,221),(27,186,234,140,111,222,134),(46,238,98,165,247,59,83),(47,248,239,60,99,84,166),(48,91,249,85,240,167,61),(49,241,92,168,250,62,86),(50,251,242,63,93,87,169),(51,94,252,88,243,170,55),(52,235,95,171,244,56,89),(53,245,236,57,96,90,163),(54,97,246,82,237,164,58)], [(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),(190,191,192,193,194,195,196,197,198),(199,200,201,202,203,204,205,206,207),(208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225),(226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243),(244,245,246,247,248,249,250,251,252)]])

60 conjugacy classes

 class 1 2 3A 3B 4A 4B 6A 6B 7A 7B 9A ··· 9F 12A 12B 12C 12D 14A 14B 18A ··· 18F 21A 21B 21C 21D 28A 28B 28C 28D 36A ··· 36L 42A 42B 42C 42D 84A ··· 84H order 1 2 3 3 4 4 6 6 7 7 9 ··· 9 12 12 12 12 14 14 18 ··· 18 21 21 21 21 28 28 28 28 36 ··· 36 42 42 42 42 84 ··· 84 size 1 1 1 1 1 1 1 1 3 3 7 ··· 7 1 1 1 1 3 3 7 ··· 7 3 3 3 3 3 3 3 3 7 ··· 7 3 3 3 3 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C4 C6 C9 C12 C18 C36 C7⋊C3 C2×C7⋊C3 C7⋊C9 C4×C7⋊C3 C2×C7⋊C9 C4×C7⋊C9 kernel C4×C7⋊C9 C2×C7⋊C9 C84 C7⋊C9 C42 C28 C21 C14 C7 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 2 2 4 4 4 8

Matrix representation of C4×C7⋊C9 in GL4(𝔽757) generated by

 1 0 0 0 0 670 0 0 0 0 670 0 0 0 0 670
,
 1 0 0 0 0 574 575 1 0 1 0 0 0 0 1 0
,
 3 0 0 0 0 720 510 437 0 597 235 255 0 437 449 559
G:=sub<GL(4,GF(757))| [1,0,0,0,0,670,0,0,0,0,670,0,0,0,0,670],[1,0,0,0,0,574,1,0,0,575,0,1,0,1,0,0],[3,0,0,0,0,720,597,437,0,510,235,449,0,437,255,559] >;

C4×C7⋊C9 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes C_9
% in TeX

G:=Group("C4xC7:C9");
// GroupNames label

G:=SmallGroup(252,2);
// by ID

G=gap.SmallGroup(252,2);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-7,30,66,909]);
// Polycyclic

G:=Group<a,b,c|a^4=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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