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## G = C12⋊3Dic6order 288 = 25·32

### 1st semidirect product of C12 and Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12⋊3Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C32⋊2Q8 — C12⋊3Dic6
 Lower central C32 — C62 — C12⋊3Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C123Dic6
G = < a,b,c | a12=b12=1, c2=b6, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 522 in 149 conjugacy classes, 60 normal (34 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×8], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×4], Dic3 [×8], C12 [×4], C12 [×8], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C3×C6 [×3], Dic6 [×8], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C4⋊Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×5], C4×C12, C3×C4⋊C4, C2×Dic6 [×4], C322Q8 [×4], C6×Dic3 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, C122Q8, C12⋊Q8, Dic3⋊Dic3 [×2], Dic3×C12, C3×C4⋊Dic3, C12⋊Dic3, C2×C322Q8 [×2], C123Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×4], C23, D6 [×6], C2×D4, C2×Q8 [×2], Dic6 [×6], D12 [×2], C22×S3 [×2], C4⋊Q8, S32, C2×Dic6 [×3], C2×D12, S3×D4, S3×Q8, C322Q8 [×2], C2×S32, C122Q8, C12⋊Q8, S3×Dic6, S3×D12, C2×C322Q8, C123Dic6

Smallest permutation representation of C123Dic6
On 96 points
Generators in S96
(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 89 56 67 9 93 52 71 5 85 60 63)(2 88 57 66 10 92 53 70 6 96 49 62)(3 87 58 65 11 91 54 69 7 95 50 61)(4 86 59 64 12 90 55 68 8 94 51 72)(13 45 79 33 17 41 83 29 21 37 75 25)(14 44 80 32 18 40 84 28 22 48 76 36)(15 43 81 31 19 39 73 27 23 47 77 35)(16 42 82 30 20 38 74 26 24 46 78 34)
(1 76 52 18)(2 77 53 19)(3 78 54 20)(4 79 55 21)(5 80 56 22)(6 81 57 23)(7 82 58 24)(8 83 59 13)(9 84 60 14)(10 73 49 15)(11 74 50 16)(12 75 51 17)(25 94 41 64)(26 95 42 65)(27 96 43 66)(28 85 44 67)(29 86 45 68)(30 87 46 69)(31 88 47 70)(32 89 48 71)(33 90 37 72)(34 91 38 61)(35 92 39 62)(36 93 40 63)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 4 2 2 6 6 6 6 12 12 36 36 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + - - + + - + - + + - - + - + image C1 C2 C2 C2 C2 C2 S3 S3 D4 Q8 Q8 D6 D6 Dic6 D12 Dic6 S32 S3×D4 S3×Q8 C32⋊2Q8 C2×S32 S3×Dic6 S3×D12 kernel C12⋊3Dic6 Dic3⋊Dic3 Dic3×C12 C3×C4⋊Dic3 C12⋊Dic3 C2×C32⋊2Q8 C4×Dic3 C4⋊Dic3 C3×Dic3 C3×Dic3 C3×C12 C2×Dic3 C2×C12 Dic3 Dic3 C12 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 1 1 2 1 1 2 2 2 4 2 4 4 8 1 1 1 2 1 2 2

Matrix representation of C123Dic6 in GL6(𝔽13)

 3 0 0 0 0 0 0 9 0 0 0 0 0 0 6 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 10 0 0 0 0 9 0 0 0 0 0 0 0 0 10 0 0 0 0 4 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,9,0,0,0,0,0,0,6,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C123Dic6 in GAP, Magma, Sage, TeX

C_{12}\rtimes_3{\rm Dic}_6
% in TeX

G:=Group("C12:3Dic6");
// GroupNames label

G:=SmallGroup(288,566);
// by ID

G=gap.SmallGroup(288,566);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,64,219,100,1356,9414]);
// Polycyclic

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

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