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## G = C11×C3.A4order 396 = 22·32·11

### Direct product of C11 and C3.A4

Aliases: C11×C3.A4, C33.A4, C22⋊C99, (C2×C22)⋊C9, C3.(C11×A4), (C2×C66).C3, (C2×C6).C33, SmallGroup(396,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C11×C3.A4
 Chief series C1 — C22 — C2×C6 — C2×C66 — C11×C3.A4
 Lower central C22 — C11×C3.A4
 Upper central C1 — C33

Generators and relations for C11×C3.A4
G = < a,b,c,d,e | a11=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C11×C3.A4
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 77 119)(2 67 120)(3 68 121)(4 69 111)(5 70 112)(6 71 113)(7 72 114)(8 73 115)(9 74 116)(10 75 117)(11 76 118)(12 194 37)(13 195 38)(14 196 39)(15 197 40)(16 198 41)(17 188 42)(18 189 43)(19 190 44)(20 191 34)(21 192 35)(22 193 36)(23 51 60)(24 52 61)(25 53 62)(26 54 63)(27 55 64)(28 45 65)(29 46 66)(30 47 56)(31 48 57)(32 49 58)(33 50 59)(78 94 104)(79 95 105)(80 96 106)(81 97 107)(82 98 108)(83 99 109)(84 89 110)(85 90 100)(86 91 101)(87 92 102)(88 93 103)(122 143 184)(123 133 185)(124 134 186)(125 135 187)(126 136 177)(127 137 178)(128 138 179)(129 139 180)(130 140 181)(131 141 182)(132 142 183)(144 160 170)(145 161 171)(146 162 172)(147 163 173)(148 164 174)(149 165 175)(150 155 176)(151 156 166)(152 157 167)(153 158 168)(154 159 169)
(1 109)(2 110)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(67 84)(68 85)(69 86)(70 87)(71 88)(72 78)(73 79)(74 80)(75 81)(76 82)(77 83)(89 120)(90 121)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 144)(129 145)(130 146)(131 147)(132 148)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)(141 163)(142 164)(143 165)(166 186)(167 187)(168 177)(169 178)(170 179)(171 180)(172 181)(173 182)(174 183)(175 184)(176 185)
(1 109)(2 110)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 45)(23 42)(24 43)(25 44)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(33 41)(56 195)(57 196)(58 197)(59 198)(60 188)(61 189)(62 190)(63 191)(64 192)(65 193)(66 194)(67 84)(68 85)(69 86)(70 87)(71 88)(72 78)(73 79)(74 80)(75 81)(76 82)(77 83)(89 120)(90 121)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)
(1 188 122 77 42 143 119 17 184)(2 189 123 67 43 133 120 18 185)(3 190 124 68 44 134 121 19 186)(4 191 125 69 34 135 111 20 187)(5 192 126 70 35 136 112 21 177)(6 193 127 71 36 137 113 22 178)(7 194 128 72 37 138 114 12 179)(8 195 129 73 38 139 115 13 180)(9 196 130 74 39 140 116 14 181)(10 197 131 75 40 141 117 15 182)(11 198 132 76 41 142 118 16 183)(23 165 99 51 175 109 60 149 83)(24 155 89 52 176 110 61 150 84)(25 156 90 53 166 100 62 151 85)(26 157 91 54 167 101 63 152 86)(27 158 92 55 168 102 64 153 87)(28 159 93 45 169 103 65 154 88)(29 160 94 46 170 104 66 144 78)(30 161 95 47 171 105 56 145 79)(31 162 96 48 172 106 57 146 80)(32 163 97 49 173 107 58 147 81)(33 164 98 50 174 108 59 148 82)

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

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

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

132 conjugacy classes

 class 1 2 3A 3B 6A 6B 9A ··· 9F 11A ··· 11J 22A ··· 22J 33A ··· 33T 66A ··· 66T 99A ··· 99BH order 1 2 3 3 6 6 9 ··· 9 11 ··· 11 22 ··· 22 33 ··· 33 66 ··· 66 99 ··· 99 size 1 3 1 1 3 3 4 ··· 4 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 4 ··· 4

132 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C3 C9 C11 C33 C99 A4 C3.A4 C11×A4 C11×C3.A4 kernel C11×C3.A4 C2×C66 C2×C22 C3.A4 C2×C6 C22 C33 C11 C3 C1 # reps 1 2 6 10 20 60 1 2 10 20

Matrix representation of C11×C3.A4 in GL3(𝔽199) generated by

 125 0 0 0 125 0 0 0 125
,
 92 0 0 0 92 0 0 0 92
,
 198 0 0 0 198 0 0 0 1
,
 198 0 0 0 1 0 0 0 198
,
 0 1 0 0 0 1 92 0 0
G:=sub<GL(3,GF(199))| [125,0,0,0,125,0,0,0,125],[92,0,0,0,92,0,0,0,92],[198,0,0,0,198,0,0,0,1],[198,0,0,0,1,0,0,0,198],[0,0,92,1,0,0,0,1,0] >;

C11×C3.A4 in GAP, Magma, Sage, TeX

C_{11}\times C_3.A_4
% in TeX

G:=Group("C11xC3.A4");
// GroupNames label

G:=SmallGroup(396,6);
// by ID

G=gap.SmallGroup(396,6);
# by ID

G:=PCGroup([5,-3,-11,-3,-2,2,165,3963,7429]);
// Polycyclic

G:=Group<a,b,c,d,e|a^11=b^3=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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