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## G = C3×C6.11D12order 432 = 24·33

### Direct product of C3 and C6.11D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C6.11D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — C2×C6×C3⋊S3 — C3×C6.11D12
 Lower central C32 — C3×C6 — C3×C6.11D12
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×C6.11D12
G = < a,b,c,d | a3=b6=c12=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 884 in 268 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C3×C22⋊C4, C3×C3⋊S3, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C3×D6⋊C4, C6.11D12, C6×C3⋊Dic3, C3×C6×C12, C2×C6×C3⋊S3, C3×C6.11D12
Quotients:

Smallest permutation representation of C3×C6.11D12
On 144 points
Generators in S144
(1 107 134)(2 108 135)(3 97 136)(4 98 137)(5 99 138)(6 100 139)(7 101 140)(8 102 141)(9 103 142)(10 104 143)(11 105 144)(12 106 133)(13 58 111)(14 59 112)(15 60 113)(16 49 114)(17 50 115)(18 51 116)(19 52 117)(20 53 118)(21 54 119)(22 55 120)(23 56 109)(24 57 110)(25 121 78)(26 122 79)(27 123 80)(28 124 81)(29 125 82)(30 126 83)(31 127 84)(32 128 73)(33 129 74)(34 130 75)(35 131 76)(36 132 77)(37 87 68)(38 88 69)(39 89 70)(40 90 71)(41 91 72)(42 92 61)(43 93 62)(44 94 63)(45 95 64)(46 96 65)(47 85 66)(48 86 67)
(1 55 142 18 99 112)(2 56 143 19 100 113)(3 57 144 20 101 114)(4 58 133 21 102 115)(5 59 134 22 103 116)(6 60 135 23 104 117)(7 49 136 24 105 118)(8 50 137 13 106 119)(9 51 138 14 107 120)(10 52 139 15 108 109)(11 53 140 16 97 110)(12 54 141 17 98 111)(25 88 129 65 82 42)(26 89 130 66 83 43)(27 90 131 67 84 44)(28 91 132 68 73 45)(29 92 121 69 74 46)(30 93 122 70 75 47)(31 94 123 71 76 48)(32 95 124 72 77 37)(33 96 125 61 78 38)(34 85 126 62 79 39)(35 86 127 63 80 40)(36 87 128 64 81 41)
(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)
(1 63 18 35)(2 34 19 62)(3 61 20 33)(4 32 21 72)(5 71 22 31)(6 30 23 70)(7 69 24 29)(8 28 13 68)(9 67 14 27)(10 26 15 66)(11 65 16 25)(12 36 17 64)(37 102 124 58)(38 57 125 101)(39 100 126 56)(40 55 127 99)(41 98 128 54)(42 53 129 97)(43 108 130 52)(44 51 131 107)(45 106 132 50)(46 49 121 105)(47 104 122 60)(48 59 123 103)(73 119 91 137)(74 136 92 118)(75 117 93 135)(76 134 94 116)(77 115 95 133)(78 144 96 114)(79 113 85 143)(80 142 86 112)(81 111 87 141)(82 140 88 110)(83 109 89 139)(84 138 90 120)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3N 4A 4B 4C 4D 6A ··· 6F 6G ··· 6AP 6AQ 6AR 6AS 6AT 12A ··· 12AZ 12BA 12BB 12BC 12BD order 1 2 2 2 2 2 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 18 18 1 1 2 ··· 2 2 2 18 18 1 ··· 1 2 ··· 2 18 18 18 18 2 ··· 2 18 18 18 18

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D6 C3×S3 C4×S3 D12 C3⋊D4 C3×D4 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 kernel C3×C6.11D12 C6×C3⋊Dic3 C3×C6×C12 C2×C6×C3⋊S3 C6.11D12 C6×C3⋊S3 C2×C3⋊Dic3 C6×C12 C22×C3⋊S3 C2×C3⋊S3 C6×C12 C32×C6 C62 C2×C12 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 1 1 1 2 4 2 2 2 8 4 2 4 8 8 8 8 4 8 16 16 16

Matrix representation of C3×C6.11D12 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 3
,
 4 0 0 0 0 0 0 10 0 0 0 0 0 0 4 0 0 0 0 0 0 10 0 0 0 0 0 0 9 0 0 0 0 0 0 3
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 9
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 9 0 0 0 0 3 0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,9,0] >;

C3×C6.11D12 in GAP, Magma, Sage, TeX

C_3\times C_6._{11}D_{12}
% in TeX

G:=Group("C3xC6.11D12");
// GroupNames label

G:=SmallGroup(432,490);
// by ID

G=gap.SmallGroup(432,490);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations

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