Copied to
clipboard

## G = C3×C6×D11order 396 = 22·32·11

### Direct product of C3×C6 and D11

Aliases: C3×C6×D11, C11⋊C62, C662C6, C22⋊(C3×C6), (C3×C66)⋊3C2, C333(C2×C6), (C3×C33)⋊8C22, SmallGroup(396,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C3×C6×D11
 Chief series C1 — C11 — C33 — C3×C33 — C32×D11 — C3×C6×D11
 Lower central C11 — C3×C6×D11
 Upper central C1 — C3×C6

Generators and relations for C3×C6×D11
G = < a,b,c,d | a3=b6=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 240 in 60 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C6, C6, C32, C11, C2×C6, C3×C6, C3×C6, D11, C22, C33, C62, D22, C3×D11, C66, C3×C33, C6×D11, C32×D11, C3×C66, C3×C6×D11
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, D11, C62, D22, C3×D11, C6×D11, C32×D11, C3×C6×D11

Smallest permutation representation of C3×C6×D11
On 198 points
Generators in S198
(1 76 43)(2 77 44)(3 67 34)(4 68 35)(5 69 36)(6 70 37)(7 71 38)(8 72 39)(9 73 40)(10 74 41)(11 75 42)(12 78 45)(13 79 46)(14 80 47)(15 81 48)(16 82 49)(17 83 50)(18 84 51)(19 85 52)(20 86 53)(21 87 54)(22 88 55)(23 89 56)(24 90 57)(25 91 58)(26 92 59)(27 93 60)(28 94 61)(29 95 62)(30 96 63)(31 97 64)(32 98 65)(33 99 66)(100 166 133)(101 167 134)(102 168 135)(103 169 136)(104 170 137)(105 171 138)(106 172 139)(107 173 140)(108 174 141)(109 175 142)(110 176 143)(111 177 144)(112 178 145)(113 179 146)(114 180 147)(115 181 148)(116 182 149)(117 183 150)(118 184 151)(119 185 152)(120 186 153)(121 187 154)(122 188 155)(123 189 156)(124 190 157)(125 191 158)(126 192 159)(127 193 160)(128 194 161)(129 195 162)(130 196 163)(131 197 164)(132 198 165)
(1 120 32 109 21 131)(2 121 33 110 22 132)(3 111 23 100 12 122)(4 112 24 101 13 123)(5 113 25 102 14 124)(6 114 26 103 15 125)(7 115 27 104 16 126)(8 116 28 105 17 127)(9 117 29 106 18 128)(10 118 30 107 19 129)(11 119 31 108 20 130)(34 144 56 133 45 155)(35 145 57 134 46 156)(36 146 58 135 47 157)(37 147 59 136 48 158)(38 148 60 137 49 159)(39 149 61 138 50 160)(40 150 62 139 51 161)(41 151 63 140 52 162)(42 152 64 141 53 163)(43 153 65 142 54 164)(44 154 66 143 55 165)(67 177 89 166 78 188)(68 178 90 167 79 189)(69 179 91 168 80 190)(70 180 92 169 81 191)(71 181 93 170 82 192)(72 182 94 171 83 193)(73 183 95 172 84 194)(74 184 96 173 85 195)(75 185 97 174 86 196)(76 186 98 175 87 197)(77 187 99 176 88 198)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)(89 95)(90 94)(91 93)(96 99)(97 98)(100 106)(101 105)(102 104)(107 110)(108 109)(111 117)(112 116)(113 115)(118 121)(119 120)(122 128)(123 127)(124 126)(129 132)(130 131)(133 139)(134 138)(135 137)(140 143)(141 142)(144 150)(145 149)(146 148)(151 154)(152 153)(155 161)(156 160)(157 159)(162 165)(163 164)(166 172)(167 171)(168 170)(173 176)(174 175)(177 183)(178 182)(179 181)(184 187)(185 186)(188 194)(189 193)(190 192)(195 198)(196 197)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 6A ··· 6H 6I ··· 6X 11A ··· 11E 22A ··· 22E 33A ··· 33AN 66A ··· 66AN order 1 2 2 2 3 ··· 3 6 ··· 6 6 ··· 6 11 ··· 11 22 ··· 22 33 ··· 33 66 ··· 66 size 1 1 11 11 1 ··· 1 1 ··· 1 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D11 D22 C3×D11 C6×D11 kernel C3×C6×D11 C32×D11 C3×C66 C6×D11 C3×D11 C66 C3×C6 C32 C6 C3 # reps 1 2 1 8 16 8 5 5 40 40

Matrix representation of C3×C6×D11 in GL4(𝔽67) generated by

 37 0 0 0 0 1 0 0 0 0 37 0 0 0 0 37
,
 1 0 0 0 0 66 0 0 0 0 37 0 0 0 0 37
,
 1 0 0 0 0 1 0 0 0 0 66 1 0 0 30 36
,
 66 0 0 0 0 66 0 0 0 0 66 0 0 0 30 1
G:=sub<GL(4,GF(67))| [37,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,0,66,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,0,1,0,0,0,0,66,30,0,0,1,36],[66,0,0,0,0,66,0,0,0,0,66,30,0,0,0,1] >;

C3×C6×D11 in GAP, Magma, Sage, TeX

C_3\times C_6\times D_{11}
% in TeX

G:=Group("C3xC6xD11");
// GroupNames label

G:=SmallGroup(396,25);
// by ID

G=gap.SmallGroup(396,25);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-11,9004]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽