Copied to
clipboard

G = C62.11C32order 324 = 22·34

2nd non-split extension by C62 of C32 acting via C32/C3=C3

Aliases: C62.11C32, C3.A4⋊C9, C3.4(C9×A4), (C3×C9).1A4, C3.1(C9⋊A4), C221(C9⋊C9), (C6×C18).4C3, C32.16(C3×A4), C3.1(C32.A4), (C2×C6).33- 1+2, (C2×C6).3(C3×C9), (C3×C3.A4).3C3, SmallGroup(324,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C62.11C32
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C62.11C32
 Lower central C22 — C2×C6 — C62.11C32
 Upper central C1 — C32 — C3×C9

Generators and relations for C62.11C32
G = < a,b,c,d | a6=b6=1, c3=b2, d3=a2, ab=ba, cac-1=ab3, ad=da, cbc-1=a3b4, bd=db, dcd-1=b2c >

Smallest permutation representation of C62.11C32
On 162 points
Generators in S162
```(1 149 89)(2 30 90 123 150 54)(3 31 82 124 151 46)(4 152 83)(5 33 84 126 153 48)(6 34 85 118 145 49)(7 146 86)(8 36 87 120 147 51)(9 28 88 121 148 52)(10 63 111 72 91 132)(11 92 112)(12 56 113 65 93 134)(13 57 114 66 94 135)(14 95 115)(15 59 116 68 96 128)(16 60 117 69 97 129)(17 98 109)(18 62 110 71 99 131)(19 139 37 79 106 154)(20 107 38)(21 141 39 81 108 156)(22 142 40 73 100 157)(23 101 41)(24 144 42 75 102 159)(25 136 43 76 103 160)(26 104 44)(27 138 45 78 105 162)(29 53 122)(32 47 125)(35 50 119)(55 133 64)(58 127 67)(61 130 70)(74 143 158)(77 137 161)(80 140 155)
(1 119 4 122 7 125)(2 8 5)(3 121 6 124 9 118)(10 69 13 72 16 66)(11 70 14 64 17 67)(12 18 15)(19 76 22 79 25 73)(20 77 23 80 26 74)(21 27 24)(28 145 31 148 34 151)(29 146 32 149 35 152)(30 36 33)(37 160 40 154 43 157)(38 161 41 155 44 158)(39 45 42)(46 88 49 82 52 85)(47 89 50 83 53 86)(48 54 51)(55 98 58 92 61 95)(56 62 59)(57 91 60 94 63 97)(65 71 68)(75 81 78)(84 90 87)(93 99 96)(100 139 103 142 106 136)(101 140 104 143 107 137)(102 108 105)(109 127 112 130 115 133)(110 116 113)(111 129 114 132 117 135)(120 126 123)(128 134 131)(138 144 141)(147 153 150)(156 162 159)
(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)
(1 133 44 89 55 104 149 64 26)(2 131 39 90 62 108 150 71 21)(3 129 43 82 60 103 151 69 25)(4 127 38 83 58 107 152 67 20)(5 134 42 84 56 102 153 65 24)(6 132 37 85 63 106 145 72 19)(7 130 41 86 61 101 146 70 23)(8 128 45 87 59 105 147 68 27)(9 135 40 88 57 100 148 66 22)(10 79 118 111 154 49 91 139 34)(11 77 122 112 161 53 92 137 29)(12 75 126 113 159 48 93 144 33)(13 73 121 114 157 52 94 142 28)(14 80 125 115 155 47 95 140 32)(15 78 120 116 162 51 96 138 36)(16 76 124 117 160 46 97 136 31)(17 74 119 109 158 50 98 143 35)(18 81 123 110 156 54 99 141 30)```

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

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

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

60 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 9A ··· 9F 9G ··· 9X 18A ··· 18R order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 3 ··· 3 3 ··· 3 12 ··· 12 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 C9 A4 3- 1+2 C3×A4 C9×A4 C9⋊A4 C32.A4 kernel C62.11C32 C3×C3.A4 C6×C18 C3.A4 C3×C9 C2×C6 C32 C3 C3 C3 # reps 1 6 2 18 1 6 2 6 12 6

Matrix representation of C62.11C32 in GL4(𝔽19) generated by

 11 0 0 0 0 1 0 0 0 18 18 0 0 18 0 18
,
 1 0 0 0 0 8 0 0 0 0 8 0 0 11 0 11
,
 7 0 0 0 0 18 17 0 0 1 1 1 0 16 1 0
,
 4 0 0 0 0 1 0 0 0 3 7 0 0 5 0 11
`G:=sub<GL(4,GF(19))| [11,0,0,0,0,1,18,18,0,0,18,0,0,0,0,18],[1,0,0,0,0,8,0,11,0,0,8,0,0,0,0,11],[7,0,0,0,0,18,1,16,0,17,1,1,0,0,1,0],[4,0,0,0,0,1,3,5,0,0,7,0,0,0,0,11] >;`

C62.11C32 in GAP, Magma, Sage, TeX

`C_6^2._{11}C_3^2`
`% in TeX`

`G:=Group("C6^2.11C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,47);`
`// by ID`

`G=gap.SmallGroup(324,47);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,361,43,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^3=b^2,d^3=a^2,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=b^2*c>;`
`// generators/relations`

Export

׿
×
𝔽