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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 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×5], C12 [×4], C12 [×3], C2×C6 [×2], C2×C6, C4⋊C4 [×2], C2×C8, C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×2], C4.Q8, C3×Dic3, C3⋊Dic3, C3×C12 [×2], C62, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3 [×3], C3×C4⋊C4, C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C2×C3⋊Dic3, C6×C12, C12.Q8, C8⋊Dic3, C6×C3⋊C8, C3×C4⋊Dic3, C12⋊Dic3, C12.Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, Dic3 [×2], D6 [×2], C4⋊C4, SD16 [×2], Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C4.Q8, S32, C24⋊C2 [×2], 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 74 5 78 9 82 13 86 17 90 21 94)(2 83 22 79 18 75 14 95 10 91 6 87)(3 76 7 80 11 84 15 88 19 92 23 96)(4 85 24 81 20 77 16 73 12 93 8 89)(25 67 29 71 33 51 37 55 41 59 45 63)(26 52 46 72 42 68 38 64 34 60 30 56)(27 69 31 49 35 53 39 57 43 61 47 65)(28 54 48 50 44 70 40 66 36 62 32 58)
(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 30 96 70)(2 41 73 57)(3 28 74 68)(4 39 75 55)(5 26 76 66)(6 37 77 53)(7 48 78 64)(8 35 79 51)(9 46 80 62)(10 33 81 49)(11 44 82 60)(12 31 83 71)(13 42 84 58)(14 29 85 69)(15 40 86 56)(16 27 87 67)(17 38 88 54)(18 25 89 65)(19 36 90 52)(20 47 91 63)(21 34 92 50)(22 45 93 61)(23 32 94 72)(24 43 95 59)

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

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

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

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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