Copied to
clipboard

G = C2×Dic3.D6order 288 = 25·32

Direct product of C2 and Dic3.D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×Dic3.D6, Dic621D6, C62.131C23, C61(S3×Q8), C6.6(S3×C23), (C3×C6).6C24, (C2×Dic6)⋊14S3, (C6×Dic6)⋊20C2, (C2×C12).168D6, C323(C22×Q8), (C2×Dic3).88D6, (C6×C12).161C22, (C3×C12).115C23, C12.108(C22×S3), (C3×Dic6)⋊26C22, C322Q810C22, C3⋊Dic3.37C23, (C3×Dic3).5C23, Dic3.4(C22×S3), C6.D6.8C22, (C6×Dic3).46C22, C31(C2×S3×Q8), C4.79(C2×S32), C3⋊S32(C2×Q8), (C3×C6)⋊3(C2×Q8), (C2×C4).123S32, (C2×C3⋊S3)⋊10Q8, C2.9(C22×S32), C22.61(C2×S32), (C2×C322Q8)⋊15C2, (C4×C3⋊S3).76C22, (C2×C3⋊S3).41C23, (C2×C6.D6).5C2, (C2×C6).148(C22×S3), (C22×C3⋊S3).100C22, (C2×C3⋊Dic3).178C22, (C2×C4×C3⋊S3).9C2, SmallGroup(288,947)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×Dic3.D6
C1C3C32C3×C6C3×Dic3C6.D6C2×C6.D6 — C2×Dic3.D6
C32C3×C6 — C2×Dic3.D6
C1C22C2×C4

Generators and relations for C2×Dic3.D6
 G = < a,b,c,d,e | a2=b6=1, c2=d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=b3d5 >

Subgroups: 1106 in 339 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×10], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×17], Q8 [×16], C23, C32, Dic3 [×8], Dic3 [×6], C12 [×4], C12 [×10], D6 [×18], C2×C6 [×2], C2×C6, C22×C4 [×3], C2×Q8 [×12], C3⋊S3 [×4], C3×C6, C3×C6 [×2], Dic6 [×8], Dic6 [×16], C4×S3 [×28], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×8], C22×S3 [×3], C22×Q8, C3×Dic3 [×8], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, C2×Dic6 [×2], C2×Dic6 [×4], S3×C2×C4 [×7], S3×Q8 [×16], C6×Q8 [×2], C6.D6 [×8], C322Q8 [×8], C3×Dic6 [×8], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×S3×Q8 [×2], Dic3.D6 [×8], C2×C6.D6 [×2], C2×C322Q8 [×2], C6×Dic6 [×2], C2×C4×C3⋊S3, C2×Dic3.D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], Q8 [×4], C23 [×15], D6 [×14], C2×Q8 [×6], C24, C22×S3 [×14], C22×Q8, S32, S3×Q8 [×4], S3×C23 [×2], C2×S32 [×3], C2×S3×Q8 [×2], Dic3.D6 [×2], C22×S32, C2×Dic3.D6

Smallest permutation representation of C2×Dic3.D6
On 48 points
Generators in S48
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 23 21 19 17 15)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 39 41 43 45 47)(38 40 42 44 46 48)
(1 30 7 36)(2 25 8 31)(3 32 9 26)(4 27 10 33)(5 34 11 28)(6 29 12 35)(13 40 19 46)(14 47 20 41)(15 42 21 48)(16 37 22 43)(17 44 23 38)(18 39 24 45)
(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 29 7 35)(2 28 8 34)(3 27 9 33)(4 26 10 32)(5 25 11 31)(6 36 12 30)(13 47 19 41)(14 46 20 40)(15 45 21 39)(16 44 22 38)(17 43 23 37)(18 42 24 48)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C···4J4K4L6A···6F6G6H6I12A···12H12I···12P
order12222222333444···4446···666612···1212···12
size11119999224226···618182···24444···412···12

48 irreducible representations

dim1111112222244444
type+++++++-++++-++
imageC1C2C2C2C2C2S3Q8D6D6D6S32S3×Q8C2×S32C2×S32Dic3.D6
kernelC2×Dic3.D6Dic3.D6C2×C6.D6C2×C322Q8C6×Dic6C2×C4×C3⋊S3C2×Dic6C2×C3⋊S3Dic6C2×Dic3C2×C12C2×C4C6C4C22C2
# reps1822212484214214

Matrix representation of C2×Dic3.D6 in GL6(𝔽13)

1200000
0120000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
0000121
0000120
,
100000
010000
008000
001500
000001
000010
,
12120000
100000
0012300
008100
000010
000001
,
12120000
010000
0051100
000800
000010
000001

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

C2×Dic3.D6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3.D_6
% in TeX

G:=Group("C2xDic3.D6");
// GroupNames label

G:=SmallGroup(288,947);
// by ID

G=gap.SmallGroup(288,947);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,675,346,80,1356,9414]);
// Polycyclic

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

׿
×
𝔽