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G = C12⋊Dic6order 288 = 25·32

1st semidirect product of C12 and Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial

Aliases: C121Dic6, C62.89C23, (C3×C12)⋊4Q8, C33(C12⋊Q8), C3⋊Dic37Q8, C6.44(S3×D4), C328(C4⋊Q8), C6.13(S3×Q8), (C2×C12).143D6, C42(C322Q8), C4⋊Dic3.13S3, C3⋊Dic3.45D4, C6.23(C2×Dic6), (C2×Dic3).37D6, C2.19(D6⋊D6), (C6×C12).111C22, C62.C22.6C2, (C6×Dic3).21C22, C2.13(Dic3.D6), (C2×C4).121S32, (C3×C6).60(C2×D4), (C3×C6).35(C2×Q8), C22.126(C2×S32), (C4×C3⋊Dic3).4C2, C2.7(C2×C322Q8), (C3×C4⋊Dic3).17C2, (C2×C322Q8).4C2, (C2×C6).108(C22×S3), (C2×C3⋊Dic3).141C22, SmallGroup(288,567)

Series: Derived Chief Lower central Upper central

C1C62 — C12⋊Dic6
C1C3C32C3×C6C62C6×Dic3C2×C322Q8 — C12⋊Dic6
C32C62 — C12⋊Dic6
C1C22C2×C4

Generators and relations for C12⋊Dic6
 G = < a,b,c | a12=b12=1, c2=b6, bab-1=a-1, cac-1=a7, cbc-1=b-1 >

Subgroups: 522 in 151 conjugacy classes, 56 normal (16 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 [×16], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C3×C6 [×3], Dic6 [×8], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C4⋊Q8, C3×Dic3 [×4], C3⋊Dic3 [×4], C3×C12 [×2], C62, C4×Dic3 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×4], C322Q8 [×4], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C12⋊Q8 [×2], C62.C22 [×2], C3×C4⋊Dic3 [×2], C4×C3⋊Dic3, C2×C322Q8 [×2], C12⋊Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×4], C23, D6 [×6], C2×D4, C2×Q8 [×2], Dic6 [×4], C22×S3 [×2], C4⋊Q8, S32, C2×Dic6 [×2], S3×D4 [×2], S3×Q8 [×2], C322Q8 [×2], C2×S32, C12⋊Q8 [×2], Dic3.D6, D6⋊D6, C2×C322Q8, C12⋊Dic6

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

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

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

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

42 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H4I4J6A···6F6G6H6I12A···12H12I···12P
order122233344444444446···666612···1212···12
size11112242212121212181818182···24444···412···12

42 irreducible representations

dim1111122222224444444
type+++++++--++-++--+
imageC1C2C2C2C2S3D4Q8Q8D6D6Dic6S32S3×D4S3×Q8C322Q8C2×S32Dic3.D6D6⋊D6
kernelC12⋊Dic6C62.C22C3×C4⋊Dic3C4×C3⋊Dic3C2×C322Q8C4⋊Dic3C3⋊Dic3C3⋊Dic3C3×C12C2×Dic3C2×C12C12C2×C4C6C6C4C22C2C2
# reps1221222224281222122

Matrix representation of C12⋊Dic6 in GL6(𝔽13)

860000
050000
00121200
001000
000010
000001
,
250000
12110000
0012000
001100
0000610
000033
,
1020000
830000
0012000
0001200
000008
000080

G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,6,5,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,12,0,0,0,0,5,11,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,6,3,0,0,0,0,10,3],[10,8,0,0,0,0,2,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;

C12⋊Dic6 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,567);
// by ID

G=gap.SmallGroup(288,567);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,64,422,219,100,1356,9414]);
// Polycyclic

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

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