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G = C3xQ8.D6order 288 = 25·32

Direct product of C3 and Q8.D6

direct product, non-abelian, soluble

Aliases: C3xQ8.D6, GL2(F3):1C6, CSU2(F3):1C6, C2.7(C6xS4), (C2xC6).5S4, (C6xQ8):4S3, C6.44(C2xS4), Q8.2(S3xC6), C22.2(C3xS4), (C3xQ8).20D6, (C2xSL2(F3)):3C6, (C6xSL2(F3)):7C2, (C3xGL2(F3)):5C2, (C3xCSU2(F3)):5C2, SL2(F3).2(C2xC6), (C3xSL2(F3)).14C22, (C2xQ8):2(C3xS3), SmallGroup(288,901)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — C3xQ8.D6
C1C2Q8SL2(F3)C3xSL2(F3)C3xGL2(F3) — C3xQ8.D6
SL2(F3) — C3xQ8.D6
C1C6C2xC6

Generators and relations for C3xQ8.D6
 G = < a,b,c,d,e | a3=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, dbd-1=bc, dcd-1=b, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 310 in 85 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C8, C2xC4, D4, Q8, Q8, C32, Dic3, C12, D6, C2xC6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3xS3, C3xC6, C24, SL2(F3), SL2(F3), C3:D4, C2xC12, C3xD4, C3xQ8, C3xQ8, C8.C22, C3xDic3, S3xC6, C62, C3xM4(2), C3xSD16, C3xQ16, CSU2(F3), GL2(F3), C2xSL2(F3), C2xSL2(F3), C6xQ8, C3xC4oD4, C3xSL2(F3), C3xC3:D4, C3xC8.C22, Q8.D6, C3xCSU2(F3), C3xGL2(F3), C6xSL2(F3), C3xQ8.D6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C3xS3, S4, S3xC6, C2xS4, C3xS4, Q8.D6, C6xS4, C3xQ8.D6

Smallest permutation representation of C3xQ8.D6
On 48 points
Generators in S48
(1 8 6)(2 7 5)(3 12 10)(4 11 9)(13 21 33)(14 22 34)(15 23 35)(16 24 36)(17 19 31)(18 20 32)(25 45 38)(26 46 39)(27 47 40)(28 48 41)(29 43 42)(30 44 37)
(1 32 11 41)(2 35 12 38)(3 45 5 23)(4 48 6 20)(7 15 10 25)(8 18 9 28)(13 27 29 17)(14 16 30 26)(19 21 47 43)(22 24 44 46)(31 33 40 42)(34 36 37 39)
(1 36 11 39)(2 33 12 42)(3 43 5 21)(4 46 6 24)(7 13 10 29)(8 16 9 26)(14 28 30 18)(15 17 25 27)(19 45 47 23)(20 22 48 44)(31 38 40 35)(32 34 41 37)
(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)
(1 2 11 12)(3 6 5 4)(7 9 10 8)(13 14 29 30)(15 18 25 28)(16 27 26 17)(19 24 47 46)(20 45 48 23)(21 22 43 44)(31 36 40 39)(32 38 41 35)(33 34 42 37)

G:=sub<Sym(48)| (1,8,6)(2,7,5)(3,12,10)(4,11,9)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32)(25,45,38)(26,46,39)(27,47,40)(28,48,41)(29,43,42)(30,44,37), (1,32,11,41)(2,35,12,38)(3,45,5,23)(4,48,6,20)(7,15,10,25)(8,18,9,28)(13,27,29,17)(14,16,30,26)(19,21,47,43)(22,24,44,46)(31,33,40,42)(34,36,37,39), (1,36,11,39)(2,33,12,42)(3,43,5,21)(4,46,6,24)(7,13,10,29)(8,16,9,26)(14,28,30,18)(15,17,25,27)(19,45,47,23)(20,22,48,44)(31,38,40,35)(32,34,41,37), (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), (1,2,11,12)(3,6,5,4)(7,9,10,8)(13,14,29,30)(15,18,25,28)(16,27,26,17)(19,24,47,46)(20,45,48,23)(21,22,43,44)(31,36,40,39)(32,38,41,35)(33,34,42,37)>;

G:=Group( (1,8,6)(2,7,5)(3,12,10)(4,11,9)(13,21,33)(14,22,34)(15,23,35)(16,24,36)(17,19,31)(18,20,32)(25,45,38)(26,46,39)(27,47,40)(28,48,41)(29,43,42)(30,44,37), (1,32,11,41)(2,35,12,38)(3,45,5,23)(4,48,6,20)(7,15,10,25)(8,18,9,28)(13,27,29,17)(14,16,30,26)(19,21,47,43)(22,24,44,46)(31,33,40,42)(34,36,37,39), (1,36,11,39)(2,33,12,42)(3,43,5,21)(4,46,6,24)(7,13,10,29)(8,16,9,26)(14,28,30,18)(15,17,25,27)(19,45,47,23)(20,22,48,44)(31,38,40,35)(32,34,41,37), (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), (1,2,11,12)(3,6,5,4)(7,9,10,8)(13,14,29,30)(15,18,25,28)(16,27,26,17)(19,24,47,46)(20,45,48,23)(21,22,43,44)(31,36,40,39)(32,38,41,35)(33,34,42,37) );

G=PermutationGroup([[(1,8,6),(2,7,5),(3,12,10),(4,11,9),(13,21,33),(14,22,34),(15,23,35),(16,24,36),(17,19,31),(18,20,32),(25,45,38),(26,46,39),(27,47,40),(28,48,41),(29,43,42),(30,44,37)], [(1,32,11,41),(2,35,12,38),(3,45,5,23),(4,48,6,20),(7,15,10,25),(8,18,9,28),(13,27,29,17),(14,16,30,26),(19,21,47,43),(22,24,44,46),(31,33,40,42),(34,36,37,39)], [(1,36,11,39),(2,33,12,42),(3,43,5,21),(4,46,6,24),(7,13,10,29),(8,16,9,26),(14,28,30,18),(15,17,25,27),(19,45,47,23),(20,22,48,44),(31,38,40,35),(32,34,41,37)], [(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)], [(1,2,11,12),(3,6,5,4),(7,9,10,8),(13,14,29,30),(15,18,25,28),(16,27,26,17),(19,24,47,46),(20,45,48,23),(21,22,43,44),(31,36,40,39),(32,38,41,35),(33,34,42,37)]])

39 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C6A6B6C6D6E···6M6N6O8A8B12A12B12C12D12E12F24A24B24C24D
order12223333344466666···6668812121212121224242424
size1121211888661211228···8121212126666121212121212

39 irreducible representations

dim1111111122223333444
type++++++++-
imageC1C2C2C2C3C6C6C6S3D6C3xS3S3xC6S4C2xS4C3xS4C6xS4Q8.D6Q8.D6C3xQ8.D6
kernelC3xQ8.D6C3xCSU2(F3)C3xGL2(F3)C6xSL2(F3)Q8.D6CSU2(F3)GL2(F3)C2xSL2(F3)C6xQ8C3xQ8C2xQ8Q8C2xC6C6C22C2C3C3C1
# reps1111222211222244126

Matrix representation of C3xQ8.D6 in GL4(F7) generated by

4000
0400
0040
0004
,
5425
6151
5340
2154
,
0634
3664
5343
2124
,
4504
6643
5423
0340
,
1626
5425
5423
0340
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,6,5,2,4,1,3,1,2,5,4,5,5,1,0,4],[0,3,5,2,6,6,3,1,3,6,4,2,4,4,3,4],[4,6,5,0,5,6,4,3,0,4,2,4,4,3,3,0],[1,5,5,0,6,4,4,3,2,2,2,4,6,5,3,0] >;

C3xQ8.D6 in GAP, Magma, Sage, TeX

C_3\times Q_8.D_6
% in TeX

G:=Group("C3xQ8.D6");
// GroupNames label

G:=SmallGroup(288,901);
// by ID

G=gap.SmallGroup(288,901);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,675,2524,655,172,1517,404,285,124]);
// Polycyclic

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

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