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G = C3×C8.6D6order 288 = 25·32

Direct product of C3 and C8.6D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C8.6D6, D24.2C6, C24.54D6, C329SD32, C3⋊C163C6, C8.6(S3×C6), C24.4(C2×C6), (C3×Q16)⋊1C6, (C3×Q16)⋊5S3, Q161(C3×S3), C33(C3×SD32), C6.10(C3×D8), C12.5(C3×D4), (C3×C6).32D8, (C3×D24).4C2, (C3×C12).43D4, C6.32(D4⋊S3), (C32×Q16)⋊1C2, C12.85(C3⋊D4), (C3×C24).15C22, (C3×C3⋊C16)⋊6C2, C2.6(C3×D4⋊S3), C4.3(C3×C3⋊D4), SmallGroup(288,262)

Series: Derived Chief Lower central Upper central

C1C24 — C3×C8.6D6
C1C3C6C12C24C3×C24C3×D24 — C3×C8.6D6
C3C6C12C24 — C3×C8.6D6
C1C6C12C24C3×Q16

Generators and relations for C3×C8.6D6
 G = < a,b,c,d | a3=b8=1, c6=b4, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 202 in 61 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C32, C12 [×2], C12 [×5], D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24 [×2], C24, D12, C3×D4, C3×Q8 [×4], SD32, C3×C12, C3×C12, S3×C6, C3⋊C16, C48, D24, C3×D8, C3×Q16 [×2], C3×Q16, C3×C24, C3×D12, Q8×C32, C8.6D6, C3×SD32, C3×C3⋊C16, C3×D24, C32×Q16, C3×C8.6D6
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, SD32, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, C8.6D6, C3×SD32, C3×D4⋊S3, C3×C8.6D6

Smallest permutation representation of C3×C8.6D6
On 96 points
Generators in S96
(1 63 36)(2 64 37)(3 49 38)(4 50 39)(5 51 40)(6 52 41)(7 53 42)(8 54 43)(9 55 44)(10 56 45)(11 57 46)(12 58 47)(13 59 48)(14 60 33)(15 61 34)(16 62 35)(17 72 91)(18 73 92)(19 74 93)(20 75 94)(21 76 95)(22 77 96)(23 78 81)(24 79 82)(25 80 83)(26 65 84)(27 66 85)(28 67 86)(29 68 87)(30 69 88)(31 70 89)(32 71 90)
(1 7 13 3 9 15 5 11)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 40 46 36 42 48 38 44)(49 55 61 51 57 63 53 59)(50 56 62 52 58 64 54 60)(65 71 77 67 73 79 69 75)(66 72 78 68 74 80 70 76)(81 87 93 83 89 95 85 91)(82 88 94 84 90 96 86 92)
(1 75 55 86 36 20 9 67 63 94 44 28)(2 27 45 93 64 66 10 19 37 85 56 74)(3 73 57 84 38 18 11 65 49 92 46 26)(4 25 47 91 50 80 12 17 39 83 58 72)(5 71 59 82 40 32 13 79 51 90 48 24)(6 23 33 89 52 78 14 31 41 81 60 70)(7 69 61 96 42 30 15 77 53 88 34 22)(8 21 35 87 54 76 16 29 43 95 62 68)
(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)

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

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

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

54 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B12A12B12C12D12E12F···12M16A16B16C16D24A24B24C24D24E···24J48A···48H
order1223333344666666688121212121212···12161616162424242424···2448···48
size1124112222811222242422224448···8666622224···46···6

54 irreducible representations

dim111111112222222222224444
type++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6D8C3×S3C3⋊D4C3×D4SD32S3×C6C3×D8C3×C3⋊D4C3×SD32D4⋊S3C8.6D6C3×D4⋊S3C3×C8.6D6
kernelC3×C8.6D6C3×C3⋊C16C3×D24C32×Q16C8.6D6C3⋊C16D24C3×Q16C3×Q16C3×C12C24C3×C6Q16C12C12C32C8C6C4C3C6C3C2C1
# reps111122221112222424481224

Matrix representation of C3×C8.6D6 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
6302
1611
2223
3426
,
2654
5235
6263
1344
,
2066
5606
4641
4512
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[2,5,6,1,6,2,2,3,5,3,6,4,4,5,3,4],[2,5,4,4,0,6,6,5,6,0,4,1,6,6,1,2] >;

C3×C8.6D6 in GAP, Magma, Sage, TeX

C_3\times C_8._6D_6
% in TeX

G:=Group("C3xC8.6D6");
// GroupNames label

G:=SmallGroup(288,262);
// by ID

G=gap.SmallGroup(288,262);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,344,1011,514,192,2524,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^4,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations

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