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## G = C6.Dic6order 144 = 24·32

### 4th non-split extension by C6 of Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6.Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C6.Dic6
 Lower central C32 — C3×C6 — C6.Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C6.Dic6
G = < a,b,c | a6=b12=1, c2=a3b6, ab=ba, cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 202 in 78 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, Dic3⋊C4, C2×C3⋊Dic3, C6×C12, C6.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, C3⋊D4, C2×C3⋊S3, Dic3⋊C4, C324Q8, C4×C3⋊S3, C327D4, C6.Dic6

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

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

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

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

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 D6 Dic6 C4×S3 C3⋊D4 kernel C6.Dic6 C2×C3⋊Dic3 C6×C12 C3⋊Dic3 C2×C12 C3×C6 C3×C6 C2×C6 C6 C6 C6 # reps 1 2 1 4 4 1 1 4 8 8 8

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

 10 0 0 0 0 4 0 0 0 0 0 1 0 0 12 1
,
 8 0 0 0 0 8 0 0 0 0 2 2 0 0 11 4
,
 0 1 0 0 1 0 0 0 0 0 8 0 0 0 8 5
G:=sub<GL(4,GF(13))| [10,0,0,0,0,4,0,0,0,0,0,12,0,0,1,1],[8,0,0,0,0,8,0,0,0,0,2,11,0,0,2,4],[0,1,0,0,1,0,0,0,0,0,8,8,0,0,0,5] >;

C6.Dic6 in GAP, Magma, Sage, TeX

C_6.{\rm Dic}_6
% in TeX

G:=Group("C6.Dic6");
// GroupNames label

G:=SmallGroup(144,93);
// by ID

G=gap.SmallGroup(144,93);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,964,3461]);
// Polycyclic

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

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