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## G = C12⋊Dic3order 144 = 24·32

### 1st semidirect product of C12 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C12⋊Dic3
 Lower central C32 — C3×C6 — C12⋊Dic3
 Upper central C1 — C22 — C2×C4

Generators and relations for C12⋊Dic3
G = < a,b,c | a12=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 202 in 78 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C3⋊Dic3, C3×C12, C62, C4⋊Dic3, C2×C3⋊Dic3, C6×C12, C12⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C3⋊S3, Dic6, D12, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4⋊Dic3, C324Q8, C12⋊S3, C2×C3⋊Dic3, C12⋊Dic3

Smallest permutation representation of C12⋊Dic3
Regular action on 144 points
Generators in S144
(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 77 27 67 14 54)(2 78 28 68 15 55)(3 79 29 69 16 56)(4 80 30 70 17 57)(5 81 31 71 18 58)(6 82 32 72 19 59)(7 83 33 61 20 60)(8 84 34 62 21 49)(9 73 35 63 22 50)(10 74 36 64 23 51)(11 75 25 65 24 52)(12 76 26 66 13 53)(37 109 132 108 144 92)(38 110 121 97 133 93)(39 111 122 98 134 94)(40 112 123 99 135 95)(41 113 124 100 136 96)(42 114 125 101 137 85)(43 115 126 102 138 86)(44 116 127 103 139 87)(45 117 128 104 140 88)(46 118 129 105 141 89)(47 119 130 106 142 90)(48 120 131 107 143 91)
(1 122 67 94)(2 121 68 93)(3 132 69 92)(4 131 70 91)(5 130 71 90)(6 129 72 89)(7 128 61 88)(8 127 62 87)(9 126 63 86)(10 125 64 85)(11 124 65 96)(12 123 66 95)(13 135 76 112)(14 134 77 111)(15 133 78 110)(16 144 79 109)(17 143 80 120)(18 142 81 119)(19 141 82 118)(20 140 83 117)(21 139 84 116)(22 138 73 115)(23 137 74 114)(24 136 75 113)(25 41 52 100)(26 40 53 99)(27 39 54 98)(28 38 55 97)(29 37 56 108)(30 48 57 107)(31 47 58 106)(32 46 59 105)(33 45 60 104)(34 44 49 103)(35 43 50 102)(36 42 51 101)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6L 12A ··· 12P order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 18 18 18 18 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C4 S3 D4 Q8 Dic3 D6 Dic6 D12 kernel C12⋊Dic3 C2×C3⋊Dic3 C6×C12 C3×C12 C2×C12 C3×C6 C3×C6 C12 C2×C6 C6 C6 # reps 1 2 1 4 4 1 1 8 4 8 8

Matrix representation of C12⋊Dic3 in GL6(𝔽13)

 10 10 0 0 0 0 3 7 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 1 12 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 8 8 0 0 0 0 0 0 10 6 0 0 0 0 3 3

G:=sub<GL(6,GF(13))| [10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,5,8,0,0,0,0,0,8,0,0,0,0,0,0,10,3,0,0,0,0,6,3] >;

C12⋊Dic3 in GAP, Magma, Sage, TeX

C_{12}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C12:Dic3");
// GroupNames label

G:=SmallGroup(144,94);
// by ID

G=gap.SmallGroup(144,94);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,55,964,3461]);
// Polycyclic

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

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