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## G = C12.Dic6order 288 = 25·32

### 5th non-split extension by C12 of Dic6 acting via Dic6/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C12.Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4⋊Dic3 — C12.Dic6
 Lower central C32 — C3×C6 — C3×C12 — C12.Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C12.Dic6
G = < a,b,c | a12=1, b12=a6, c2=a9b6, bab-1=a5, cac-1=a7, cbc-1=b11 >

Subgroups: 298 in 85 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C4.Q8, C3×Dic3, C3⋊Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C3×C3⋊C8, C6×Dic3, C2×C3⋊Dic3, C6×C12, C12.Q8, C8⋊Dic3, C6×C3⋊C8, C3×C4⋊Dic3, C12⋊Dic3, C12.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, SD16, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C4.Q8, S32, C24⋊C2, Dic3⋊C4, C4⋊Dic3, D4.S3, Q82S3, S3×Dic3, C3⋊D12, C322Q8, C12.Q8, C8⋊Dic3, D12.S3, C325SD16, Dic3⋊Dic3, C12.Dic6

Smallest permutation representation of C12.Dic6
On 96 points
Generators in S96
(1 84 5 88 9 92 13 96 17 76 21 80)(2 93 22 89 18 85 14 81 10 77 6 73)(3 86 7 90 11 94 15 74 19 78 23 82)(4 95 24 91 20 87 16 83 12 79 8 75)(25 71 29 51 33 55 37 59 41 63 45 67)(26 56 46 52 42 72 38 68 34 64 30 60)(27 49 31 53 35 57 39 61 43 65 47 69)(28 58 48 54 44 50 40 70 36 66 32 62)
(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)
(1 50 82 30)(2 61 83 41)(3 72 84 28)(4 59 85 39)(5 70 86 26)(6 57 87 37)(7 68 88 48)(8 55 89 35)(9 66 90 46)(10 53 91 33)(11 64 92 44)(12 51 93 31)(13 62 94 42)(14 49 95 29)(15 60 96 40)(16 71 73 27)(17 58 74 38)(18 69 75 25)(19 56 76 36)(20 67 77 47)(21 54 78 34)(22 65 79 45)(23 52 80 32)(24 63 81 43)

G:=sub<Sym(96)| (1,84,5,88,9,92,13,96,17,76,21,80)(2,93,22,89,18,85,14,81,10,77,6,73)(3,86,7,90,11,94,15,74,19,78,23,82)(4,95,24,91,20,87,16,83,12,79,8,75)(25,71,29,51,33,55,37,59,41,63,45,67)(26,56,46,52,42,72,38,68,34,64,30,60)(27,49,31,53,35,57,39,61,43,65,47,69)(28,58,48,54,44,50,40,70,36,66,32,62), (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), (1,50,82,30)(2,61,83,41)(3,72,84,28)(4,59,85,39)(5,70,86,26)(6,57,87,37)(7,68,88,48)(8,55,89,35)(9,66,90,46)(10,53,91,33)(11,64,92,44)(12,51,93,31)(13,62,94,42)(14,49,95,29)(15,60,96,40)(16,71,73,27)(17,58,74,38)(18,69,75,25)(19,56,76,36)(20,67,77,47)(21,54,78,34)(22,65,79,45)(23,52,80,32)(24,63,81,43)>;

G:=Group( (1,84,5,88,9,92,13,96,17,76,21,80)(2,93,22,89,18,85,14,81,10,77,6,73)(3,86,7,90,11,94,15,74,19,78,23,82)(4,95,24,91,20,87,16,83,12,79,8,75)(25,71,29,51,33,55,37,59,41,63,45,67)(26,56,46,52,42,72,38,68,34,64,30,60)(27,49,31,53,35,57,39,61,43,65,47,69)(28,58,48,54,44,50,40,70,36,66,32,62), (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), (1,50,82,30)(2,61,83,41)(3,72,84,28)(4,59,85,39)(5,70,86,26)(6,57,87,37)(7,68,88,48)(8,55,89,35)(9,66,90,46)(10,53,91,33)(11,64,92,44)(12,51,93,31)(13,62,94,42)(14,49,95,29)(15,60,96,40)(16,71,73,27)(17,58,74,38)(18,69,75,25)(19,56,76,36)(20,67,77,47)(21,54,78,34)(22,65,79,45)(23,52,80,32)(24,63,81,43) );

G=PermutationGroup([[(1,84,5,88,9,92,13,96,17,76,21,80),(2,93,22,89,18,85,14,81,10,77,6,73),(3,86,7,90,11,94,15,74,19,78,23,82),(4,95,24,91,20,87,16,83,12,79,8,75),(25,71,29,51,33,55,37,59,41,63,45,67),(26,56,46,52,42,72,38,68,34,64,30,60),(27,49,31,53,35,57,39,61,43,65,47,69),(28,58,48,54,44,50,40,70,36,66,32,62)], [(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)], [(1,50,82,30),(2,61,83,41),(3,72,84,28),(4,59,85,39),(5,70,86,26),(6,57,87,37),(7,68,88,48),(8,55,89,35),(9,66,90,46),(10,53,91,33),(11,64,92,44),(12,51,93,31),(13,62,94,42),(14,49,95,29),(15,60,96,40),(16,71,73,27),(17,58,74,38),(18,69,75,25),(19,56,76,36),(20,67,77,47),(21,54,78,34),(22,65,79,45),(23,52,80,32),(24,63,81,43)]])

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 2 2 12 12 36 36 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + - + - + - + + - + - - + - + image C1 C2 C2 C2 C4 S3 S3 Q8 D4 Dic3 D6 SD16 Dic6 C4×S3 D12 C3⋊D4 C24⋊C2 S32 D4.S3 Q8⋊2S3 S3×Dic3 C32⋊2Q8 C3⋊D12 D12.S3 C32⋊5SD16 kernel C12.Dic6 C6×C3⋊C8 C3×C4⋊Dic3 C12⋊Dic3 C3×C3⋊C8 C2×C3⋊C8 C4⋊Dic3 C3×C12 C62 C3⋊C8 C2×C12 C3×C6 C12 C12 C2×C6 C2×C6 C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 4 4 2 2 2 8 1 1 1 1 1 1 2 2

Matrix representation of C12.Dic6 in GL6(𝔽73)

 72 2 0 0 0 0 72 1 0 0 0 0 0 0 0 1 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 67 12 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 59 7 0 0 0 0 66 66
,
 46 54 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 14 5 0 0 0 0 19 59

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,2,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,67,0,0,0,0,12,12,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,59,66,0,0,0,0,7,66],[46,0,0,0,0,0,54,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,14,19,0,0,0,0,5,59] >;

C12.Dic6 in GAP, Magma, Sage, TeX

C_{12}.{\rm Dic}_6
% in TeX

G:=Group("C12.Dic6");
// GroupNames label

G:=SmallGroup(288,221);
// by ID

G=gap.SmallGroup(288,221);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,36,422,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9*b^6,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=b^11>;
// generators/relations

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