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## G = C3⋊S3×C22order 396 = 22·32·11

### Direct product of C22 and C3⋊S3

Aliases: C3⋊S3×C22, C663S3, C338D6, C6⋊(S3×C11), C32(S3×C22), (C3×C66)⋊5C2, (C3×C6)⋊2C22, C323(C2×C22), (C3×C33)⋊10C22, SmallGroup(396,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C22
 Chief series C1 — C3 — C32 — C3×C33 — C11×C3⋊S3 — C3⋊S3×C22
 Lower central C32 — C3⋊S3×C22
 Upper central C1 — C22

Generators and relations for C3⋊S3×C22
G = < a,b,c,d | a22=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 156 in 60 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2, C3, C22, S3, C6, C32, C11, D6, C3⋊S3, C3×C6, C22, C22, C33, C2×C3⋊S3, C2×C22, S3×C11, C66, C3×C33, S3×C22, C11×C3⋊S3, C3×C66, C3⋊S3×C22
Quotients: C1, C2, C22, S3, C11, D6, C3⋊S3, C22, C2×C3⋊S3, C2×C22, S3×C11, S3×C22, C11×C3⋊S3, C3⋊S3×C22

Smallest permutation representation of C3⋊S3×C22
On 198 points
Generators in S198
(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)
(1 146 131)(2 147 132)(3 148 111)(4 149 112)(5 150 113)(6 151 114)(7 152 115)(8 153 116)(9 154 117)(10 133 118)(11 134 119)(12 135 120)(13 136 121)(14 137 122)(15 138 123)(16 139 124)(17 140 125)(18 141 126)(19 142 127)(20 143 128)(21 144 129)(22 145 130)(23 177 55)(24 178 56)(25 179 57)(26 180 58)(27 181 59)(28 182 60)(29 183 61)(30 184 62)(31 185 63)(32 186 64)(33 187 65)(34 188 66)(35 189 45)(36 190 46)(37 191 47)(38 192 48)(39 193 49)(40 194 50)(41 195 51)(42 196 52)(43 197 53)(44 198 54)(67 104 159)(68 105 160)(69 106 161)(70 107 162)(71 108 163)(72 109 164)(73 110 165)(74 89 166)(75 90 167)(76 91 168)(77 92 169)(78 93 170)(79 94 171)(80 95 172)(81 96 173)(82 97 174)(83 98 175)(84 99 176)(85 100 155)(86 101 156)(87 102 157)(88 103 158)
(1 28 101)(2 29 102)(3 30 103)(4 31 104)(5 32 105)(6 33 106)(7 34 107)(8 35 108)(9 36 109)(10 37 110)(11 38 89)(12 39 90)(13 40 91)(14 41 92)(15 42 93)(16 43 94)(17 44 95)(18 23 96)(19 24 97)(20 25 98)(21 26 99)(22 27 100)(45 71 116)(46 72 117)(47 73 118)(48 74 119)(49 75 120)(50 76 121)(51 77 122)(52 78 123)(53 79 124)(54 80 125)(55 81 126)(56 82 127)(57 83 128)(58 84 129)(59 85 130)(60 86 131)(61 87 132)(62 88 111)(63 67 112)(64 68 113)(65 69 114)(66 70 115)(133 191 165)(134 192 166)(135 193 167)(136 194 168)(137 195 169)(138 196 170)(139 197 171)(140 198 172)(141 177 173)(142 178 174)(143 179 175)(144 180 176)(145 181 155)(146 182 156)(147 183 157)(148 184 158)(149 185 159)(150 186 160)(151 187 161)(152 188 162)(153 189 163)(154 190 164)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 107)(24 108)(25 109)(26 110)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 174)(46 175)(47 176)(48 155)(49 156)(50 157)(51 158)(52 159)(53 160)(54 161)(55 162)(56 163)(57 164)(58 165)(59 166)(60 167)(61 168)(62 169)(63 170)(64 171)(65 172)(66 173)(67 196)(68 197)(69 198)(70 177)(71 178)(72 179)(73 180)(74 181)(75 182)(76 183)(77 184)(78 185)(79 186)(80 187)(81 188)(82 189)(83 190)(84 191)(85 192)(86 193)(87 194)(88 195)(111 137)(112 138)(113 139)(114 140)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 133)(130 134)(131 135)(132 136)

G:=sub<Sym(198)| (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), (1,146,131)(2,147,132)(3,148,111)(4,149,112)(5,150,113)(6,151,114)(7,152,115)(8,153,116)(9,154,117)(10,133,118)(11,134,119)(12,135,120)(13,136,121)(14,137,122)(15,138,123)(16,139,124)(17,140,125)(18,141,126)(19,142,127)(20,143,128)(21,144,129)(22,145,130)(23,177,55)(24,178,56)(25,179,57)(26,180,58)(27,181,59)(28,182,60)(29,183,61)(30,184,62)(31,185,63)(32,186,64)(33,187,65)(34,188,66)(35,189,45)(36,190,46)(37,191,47)(38,192,48)(39,193,49)(40,194,50)(41,195,51)(42,196,52)(43,197,53)(44,198,54)(67,104,159)(68,105,160)(69,106,161)(70,107,162)(71,108,163)(72,109,164)(73,110,165)(74,89,166)(75,90,167)(76,91,168)(77,92,169)(78,93,170)(79,94,171)(80,95,172)(81,96,173)(82,97,174)(83,98,175)(84,99,176)(85,100,155)(86,101,156)(87,102,157)(88,103,158), (1,28,101)(2,29,102)(3,30,103)(4,31,104)(5,32,105)(6,33,106)(7,34,107)(8,35,108)(9,36,109)(10,37,110)(11,38,89)(12,39,90)(13,40,91)(14,41,92)(15,42,93)(16,43,94)(17,44,95)(18,23,96)(19,24,97)(20,25,98)(21,26,99)(22,27,100)(45,71,116)(46,72,117)(47,73,118)(48,74,119)(49,75,120)(50,76,121)(51,77,122)(52,78,123)(53,79,124)(54,80,125)(55,81,126)(56,82,127)(57,83,128)(58,84,129)(59,85,130)(60,86,131)(61,87,132)(62,88,111)(63,67,112)(64,68,113)(65,69,114)(66,70,115)(133,191,165)(134,192,166)(135,193,167)(136,194,168)(137,195,169)(138,196,170)(139,197,171)(140,198,172)(141,177,173)(142,178,174)(143,179,175)(144,180,176)(145,181,155)(146,182,156)(147,183,157)(148,184,158)(149,185,159)(150,186,160)(151,187,161)(152,188,162)(153,189,163)(154,190,164), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,107)(24,108)(25,109)(26,110)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,174)(46,175)(47,176)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,170)(64,171)(65,172)(66,173)(67,196)(68,197)(69,198)(70,177)(71,178)(72,179)(73,180)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(80,187)(81,188)(82,189)(83,190)(84,191)(85,192)(86,193)(87,194)(88,195)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,133)(130,134)(131,135)(132,136)>;

G:=Group( (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), (1,146,131)(2,147,132)(3,148,111)(4,149,112)(5,150,113)(6,151,114)(7,152,115)(8,153,116)(9,154,117)(10,133,118)(11,134,119)(12,135,120)(13,136,121)(14,137,122)(15,138,123)(16,139,124)(17,140,125)(18,141,126)(19,142,127)(20,143,128)(21,144,129)(22,145,130)(23,177,55)(24,178,56)(25,179,57)(26,180,58)(27,181,59)(28,182,60)(29,183,61)(30,184,62)(31,185,63)(32,186,64)(33,187,65)(34,188,66)(35,189,45)(36,190,46)(37,191,47)(38,192,48)(39,193,49)(40,194,50)(41,195,51)(42,196,52)(43,197,53)(44,198,54)(67,104,159)(68,105,160)(69,106,161)(70,107,162)(71,108,163)(72,109,164)(73,110,165)(74,89,166)(75,90,167)(76,91,168)(77,92,169)(78,93,170)(79,94,171)(80,95,172)(81,96,173)(82,97,174)(83,98,175)(84,99,176)(85,100,155)(86,101,156)(87,102,157)(88,103,158), (1,28,101)(2,29,102)(3,30,103)(4,31,104)(5,32,105)(6,33,106)(7,34,107)(8,35,108)(9,36,109)(10,37,110)(11,38,89)(12,39,90)(13,40,91)(14,41,92)(15,42,93)(16,43,94)(17,44,95)(18,23,96)(19,24,97)(20,25,98)(21,26,99)(22,27,100)(45,71,116)(46,72,117)(47,73,118)(48,74,119)(49,75,120)(50,76,121)(51,77,122)(52,78,123)(53,79,124)(54,80,125)(55,81,126)(56,82,127)(57,83,128)(58,84,129)(59,85,130)(60,86,131)(61,87,132)(62,88,111)(63,67,112)(64,68,113)(65,69,114)(66,70,115)(133,191,165)(134,192,166)(135,193,167)(136,194,168)(137,195,169)(138,196,170)(139,197,171)(140,198,172)(141,177,173)(142,178,174)(143,179,175)(144,180,176)(145,181,155)(146,182,156)(147,183,157)(148,184,158)(149,185,159)(150,186,160)(151,187,161)(152,188,162)(153,189,163)(154,190,164), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,107)(24,108)(25,109)(26,110)(27,89)(28,90)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,174)(46,175)(47,176)(48,155)(49,156)(50,157)(51,158)(52,159)(53,160)(54,161)(55,162)(56,163)(57,164)(58,165)(59,166)(60,167)(61,168)(62,169)(63,170)(64,171)(65,172)(66,173)(67,196)(68,197)(69,198)(70,177)(71,178)(72,179)(73,180)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(80,187)(81,188)(82,189)(83,190)(84,191)(85,192)(86,193)(87,194)(88,195)(111,137)(112,138)(113,139)(114,140)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,133)(130,134)(131,135)(132,136) );

G=PermutationGroup([[(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)], [(1,146,131),(2,147,132),(3,148,111),(4,149,112),(5,150,113),(6,151,114),(7,152,115),(8,153,116),(9,154,117),(10,133,118),(11,134,119),(12,135,120),(13,136,121),(14,137,122),(15,138,123),(16,139,124),(17,140,125),(18,141,126),(19,142,127),(20,143,128),(21,144,129),(22,145,130),(23,177,55),(24,178,56),(25,179,57),(26,180,58),(27,181,59),(28,182,60),(29,183,61),(30,184,62),(31,185,63),(32,186,64),(33,187,65),(34,188,66),(35,189,45),(36,190,46),(37,191,47),(38,192,48),(39,193,49),(40,194,50),(41,195,51),(42,196,52),(43,197,53),(44,198,54),(67,104,159),(68,105,160),(69,106,161),(70,107,162),(71,108,163),(72,109,164),(73,110,165),(74,89,166),(75,90,167),(76,91,168),(77,92,169),(78,93,170),(79,94,171),(80,95,172),(81,96,173),(82,97,174),(83,98,175),(84,99,176),(85,100,155),(86,101,156),(87,102,157),(88,103,158)], [(1,28,101),(2,29,102),(3,30,103),(4,31,104),(5,32,105),(6,33,106),(7,34,107),(8,35,108),(9,36,109),(10,37,110),(11,38,89),(12,39,90),(13,40,91),(14,41,92),(15,42,93),(16,43,94),(17,44,95),(18,23,96),(19,24,97),(20,25,98),(21,26,99),(22,27,100),(45,71,116),(46,72,117),(47,73,118),(48,74,119),(49,75,120),(50,76,121),(51,77,122),(52,78,123),(53,79,124),(54,80,125),(55,81,126),(56,82,127),(57,83,128),(58,84,129),(59,85,130),(60,86,131),(61,87,132),(62,88,111),(63,67,112),(64,68,113),(65,69,114),(66,70,115),(133,191,165),(134,192,166),(135,193,167),(136,194,168),(137,195,169),(138,196,170),(139,197,171),(140,198,172),(141,177,173),(142,178,174),(143,179,175),(144,180,176),(145,181,155),(146,182,156),(147,183,157),(148,184,158),(149,185,159),(150,186,160),(151,187,161),(152,188,162),(153,189,163),(154,190,164)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,107),(24,108),(25,109),(26,110),(27,89),(28,90),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,105),(44,106),(45,174),(46,175),(47,176),(48,155),(49,156),(50,157),(51,158),(52,159),(53,160),(54,161),(55,162),(56,163),(57,164),(58,165),(59,166),(60,167),(61,168),(62,169),(63,170),(64,171),(65,172),(66,173),(67,196),(68,197),(69,198),(70,177),(71,178),(72,179),(73,180),(74,181),(75,182),(76,183),(77,184),(78,185),(79,186),(80,187),(81,188),(82,189),(83,190),(84,191),(85,192),(86,193),(87,194),(88,195),(111,137),(112,138),(113,139),(114,140),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,153),(128,154),(129,133),(130,134),(131,135),(132,136)]])

132 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 11A ··· 11J 22A ··· 22J 22K ··· 22AD 33A ··· 33AN 66A ··· 66AN order 1 2 2 2 3 3 3 3 6 6 6 6 11 ··· 11 22 ··· 22 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 9 9 2 2 2 2 2 2 2 2 1 ··· 1 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

132 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C11 C22 C22 S3 D6 S3×C11 S3×C22 kernel C3⋊S3×C22 C11×C3⋊S3 C3×C66 C2×C3⋊S3 C3⋊S3 C3×C6 C66 C33 C6 C3 # reps 1 2 1 10 20 10 4 4 40 40

Matrix representation of C3⋊S3×C22 in GL4(𝔽67) generated by

 1 0 0 0 0 1 0 0 0 0 42 0 0 0 0 42
,
 0 1 0 0 66 66 0 0 0 0 0 1 0 0 66 66
,
 66 66 0 0 1 0 0 0 0 0 0 1 0 0 66 66
,
 1 0 0 0 66 66 0 0 0 0 0 66 0 0 66 0
G:=sub<GL(4,GF(67))| [1,0,0,0,0,1,0,0,0,0,42,0,0,0,0,42],[0,66,0,0,1,66,0,0,0,0,0,66,0,0,1,66],[66,1,0,0,66,0,0,0,0,0,0,66,0,0,1,66],[1,66,0,0,0,66,0,0,0,0,0,66,0,0,66,0] >;

C3⋊S3×C22 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_{22}
% in TeX

G:=Group("C3:S3xC22");
// GroupNames label

G:=SmallGroup(396,28);
// by ID

G=gap.SmallGroup(396,28);
# by ID

G:=PCGroup([5,-2,-2,-11,-3,-3,1763,6604]);
// Polycyclic

G:=Group<a,b,c,d|a^22=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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