Copied to
clipboard

## G = C3×C6.Dic6order 432 = 24·33

### Direct product of C3 and C6.Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C6.Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — C6×C3⋊Dic3 — C3×C6.Dic6
 Lower central C32 — C3×C6 — C3×C6.Dic6
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×C6.Dic6
G = < a,b,c,d | a3=b6=c12=1, d2=b3c6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 532 in 220 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C3×C4⋊C4, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C3×C3⋊Dic3, C32×C12, C3×C62, C3×Dic3⋊C4, C6.Dic6, C6×C3⋊Dic3, C3×C6×C12, C3×C6.Dic6
Quotients:

Smallest permutation representation of C3×C6.Dic6
On 144 points
Generators in S144
(1 117 26)(2 118 27)(3 119 28)(4 120 29)(5 109 30)(6 110 31)(7 111 32)(8 112 33)(9 113 34)(10 114 35)(11 115 36)(12 116 25)(13 137 121)(14 138 122)(15 139 123)(16 140 124)(17 141 125)(18 142 126)(19 143 127)(20 144 128)(21 133 129)(22 134 130)(23 135 131)(24 136 132)(37 100 53)(38 101 54)(39 102 55)(40 103 56)(41 104 57)(42 105 58)(43 106 59)(44 107 60)(45 108 49)(46 97 50)(47 98 51)(48 99 52)(61 82 86)(62 83 87)(63 84 88)(64 73 89)(65 74 90)(66 75 91)(67 76 92)(68 77 93)(69 78 94)(70 79 95)(71 80 96)(72 81 85)
(1 67 30 96 113 84)(2 68 31 85 114 73)(3 69 32 86 115 74)(4 70 33 87 116 75)(5 71 34 88 117 76)(6 72 35 89 118 77)(7 61 36 90 119 78)(8 62 25 91 120 79)(9 63 26 92 109 80)(10 64 27 93 110 81)(11 65 28 94 111 82)(12 66 29 95 112 83)(13 45 141 100 129 57)(14 46 142 101 130 58)(15 47 143 102 131 59)(16 48 144 103 132 60)(17 37 133 104 121 49)(18 38 134 105 122 50)(19 39 135 106 123 51)(20 40 136 107 124 52)(21 41 137 108 125 53)(22 42 138 97 126 54)(23 43 139 98 127 55)(24 44 140 99 128 56)
(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)
(1 130 90 40)(2 45 91 123)(3 128 92 38)(4 43 93 121)(5 126 94 48)(6 41 95 131)(7 124 96 46)(8 39 85 129)(9 122 86 44)(10 37 87 127)(11 132 88 42)(12 47 89 125)(13 120 106 68)(14 61 107 113)(15 118 108 66)(16 71 97 111)(17 116 98 64)(18 69 99 109)(19 114 100 62)(20 67 101 119)(21 112 102 72)(22 65 103 117)(23 110 104 70)(24 63 105 115)(25 51 73 141)(26 134 74 56)(27 49 75 139)(28 144 76 54)(29 59 77 137)(30 142 78 52)(31 57 79 135)(32 140 80 50)(33 55 81 133)(34 138 82 60)(35 53 83 143)(36 136 84 58)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6AP 12A ··· 12AZ 12BA ··· 12BH order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 ··· 2 2 2 18 18 18 18 1 ··· 1 2 ··· 2 2 ··· 2 18 ··· 18

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 D6 C3×S3 Dic6 C4×S3 C3⋊D4 C3×D4 C3×Q8 S3×C6 C3×Dic6 S3×C12 C3×C3⋊D4 kernel C3×C6.Dic6 C6×C3⋊Dic3 C3×C6×C12 C6.Dic6 C3×C3⋊Dic3 C2×C3⋊Dic3 C6×C12 C3⋊Dic3 C6×C12 C32×C6 C32×C6 C62 C2×C12 C3×C6 C3×C6 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 2 1 2 4 4 2 8 4 1 1 4 8 8 8 8 2 2 8 16 16 16

Matrix representation of C3×C6.Dic6 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 12 0 0 0 0 12 0 0 0 0 4 0 0 0 0 10
,
 9 0 0 0 0 10 0 0 0 0 6 0 0 0 0 2
,
 0 1 0 0 12 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[12,0,0,0,0,12,0,0,0,0,4,0,0,0,0,10],[9,0,0,0,0,10,0,0,0,0,6,0,0,0,0,2],[0,12,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C6.Dic6 in GAP, Magma, Sage, TeX

C_3\times C_6.{\rm Dic}_6
% in TeX

G:=Group("C3xC6.Dic6");
// GroupNames label

G:=SmallGroup(432,488);
// by ID

G=gap.SmallGroup(432,488);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,365,92,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=b^3*c^6,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

׿
×
𝔽