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G = C3⋊Dic21order 252 = 22·32·7

The semidirect product of C3 and Dic21 acting via Dic21/C42=C2

metabelian, supersoluble, monomial, A-group

Aliases: C3⋊Dic21, C42.3S3, C6.3D21, C211Dic3, C323Dic7, (C3×C21)⋊3C4, C7⋊(C3⋊Dic3), C14.(C3⋊S3), C2.(C3⋊D21), (C3×C6).2D7, (C3×C42).1C2, SmallGroup(252,24)

Series: Derived Chief Lower central Upper central

C1C3×C21 — C3⋊Dic21
C1C7C21C3×C21C3×C42 — C3⋊Dic21
C3×C21 — C3⋊Dic21
C1C2

Generators and relations for C3⋊Dic21
 G = < a,b,c | a3=b42=1, c2=b21, ab=ba, cac-1=a-1, cbc-1=b-1 >

63C4
21Dic3
21Dic3
21Dic3
21Dic3
9Dic7
7C3⋊Dic3
3Dic21
3Dic21
3Dic21
3Dic21

Smallest permutation representation of C3⋊Dic21
Regular action on 252 points
Generators in S252
(1 64 100)(2 65 101)(3 66 102)(4 67 103)(5 68 104)(6 69 105)(7 70 106)(8 71 107)(9 72 108)(10 73 109)(11 74 110)(12 75 111)(13 76 112)(14 77 113)(15 78 114)(16 79 115)(17 80 116)(18 81 117)(19 82 118)(20 83 119)(21 84 120)(22 43 121)(23 44 122)(24 45 123)(25 46 124)(26 47 125)(27 48 126)(28 49 85)(29 50 86)(30 51 87)(31 52 88)(32 53 89)(33 54 90)(34 55 91)(35 56 92)(36 57 93)(37 58 94)(38 59 95)(39 60 96)(40 61 97)(41 62 98)(42 63 99)(127 237 184)(128 238 185)(129 239 186)(130 240 187)(131 241 188)(132 242 189)(133 243 190)(134 244 191)(135 245 192)(136 246 193)(137 247 194)(138 248 195)(139 249 196)(140 250 197)(141 251 198)(142 252 199)(143 211 200)(144 212 201)(145 213 202)(146 214 203)(147 215 204)(148 216 205)(149 217 206)(150 218 207)(151 219 208)(152 220 209)(153 221 210)(154 222 169)(155 223 170)(156 224 171)(157 225 172)(158 226 173)(159 227 174)(160 228 175)(161 229 176)(162 230 177)(163 231 178)(164 232 179)(165 233 180)(166 234 181)(167 235 182)(168 236 183)
(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)
(1 198 22 177)(2 197 23 176)(3 196 24 175)(4 195 25 174)(5 194 26 173)(6 193 27 172)(7 192 28 171)(8 191 29 170)(9 190 30 169)(10 189 31 210)(11 188 32 209)(12 187 33 208)(13 186 34 207)(14 185 35 206)(15 184 36 205)(16 183 37 204)(17 182 38 203)(18 181 39 202)(19 180 40 201)(20 179 41 200)(21 178 42 199)(43 230 64 251)(44 229 65 250)(45 228 66 249)(46 227 67 248)(47 226 68 247)(48 225 69 246)(49 224 70 245)(50 223 71 244)(51 222 72 243)(52 221 73 242)(53 220 74 241)(54 219 75 240)(55 218 76 239)(56 217 77 238)(57 216 78 237)(58 215 79 236)(59 214 80 235)(60 213 81 234)(61 212 82 233)(62 211 83 232)(63 252 84 231)(85 156 106 135)(86 155 107 134)(87 154 108 133)(88 153 109 132)(89 152 110 131)(90 151 111 130)(91 150 112 129)(92 149 113 128)(93 148 114 127)(94 147 115 168)(95 146 116 167)(96 145 117 166)(97 144 118 165)(98 143 119 164)(99 142 120 163)(100 141 121 162)(101 140 122 161)(102 139 123 160)(103 138 124 159)(104 137 125 158)(105 136 126 157)

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

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

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

66 conjugacy classes

class 1  2 3A3B3C3D4A4B6A6B6C6D7A7B7C14A14B14C21A···21X42A···42X
order12333344666677714141421···2142···42
size112222636322222222222···22···2

66 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D7Dic7D21Dic21
kernelC3⋊Dic21C3×C42C3×C21C42C21C3×C6C32C6C3
# reps11244332424

Matrix representation of C3⋊Dic21 in GL4(𝔽337) generated by

23623000
10710000
00100107
00230236
,
10710100
2365900
000336
001194
,
8217500
10625500
00255162
0023182
G:=sub<GL(4,GF(337))| [236,107,0,0,230,100,0,0,0,0,100,230,0,0,107,236],[107,236,0,0,101,59,0,0,0,0,0,1,0,0,336,194],[82,106,0,0,175,255,0,0,0,0,255,231,0,0,162,82] >;

C3⋊Dic21 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_{21}
% in TeX

G:=Group("C3:Dic21");
// GroupNames label

G:=SmallGroup(252,24);
// by ID

G=gap.SmallGroup(252,24);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,10,122,483,5404]);
// Polycyclic

G:=Group<a,b,c|a^3=b^42=1,c^2=b^21,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3⋊Dic21 in TeX

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