Copied to
clipboard

## G = C2×D6.6D6order 288 = 25·32

### Direct product of C2 and D6.6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×D6.6D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C3⋊D12 — C2×C3⋊D12 — C2×D6.6D6
 Lower central C32 — C3×C6 — C2×D6.6D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D6.6D6
G = < a,b,c,d,e | a2=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d5 >

Subgroups: 1346 in 355 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×14], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×6], C12 [×4], C12 [×8], D6 [×2], D6 [×26], C2×C6 [×2], C2×C6 [×5], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3⋊S3 [×4], C3×C6, C3×C6 [×2], Dic6 [×4], C4×S3 [×4], C4×S3 [×16], D12 [×20], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×8], C3×Q8 [×4], C22×S3, C22×S3 [×6], C22×C6, C2×C4○D4, C3×Dic3 [×6], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C2×Dic6, S3×C2×C4, S3×C2×C4 [×4], C2×D12 [×5], C4○D12 [×8], Q83S3 [×8], C2×C3⋊D4 [×2], C22×C12, C6×Q8, C6.D6 [×8], C3⋊D12 [×8], C3×Dic6 [×4], S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C12⋊S3 [×4], C6×C12, S3×C2×C6, C22×C3⋊S3 [×2], C2×C4○D12, C2×Q83S3, D6.6D6 [×8], C2×C6.D6 [×2], C2×C3⋊D12 [×2], C6×Dic6, S3×C2×C12, C2×C12⋊S3, C2×D6.6D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, C4○D12 [×2], Q83S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×C4○D12, C2×Q83S3, D6.6D6 [×2], C22×S32, C2×D6.6D6

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R order 1 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 6 6 18 18 18 18 2 2 4 2 2 3 3 3 3 6 6 6 6 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 C4○D4 C4○D12 S32 Q8⋊3S3 C2×S32 C2×S32 D6.6D6 kernel C2×D6.6D6 D6.6D6 C2×C6.D6 C2×C3⋊D12 C6×Dic6 S3×C2×C12 C2×C12⋊S3 C2×Dic6 S3×C2×C4 Dic6 C4×S3 C2×Dic3 C2×C12 C22×S3 C3×C6 C6 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 1 4 4 3 2 1 4 8 1 2 2 1 4

Matrix representation of C2×D6.6D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 1 0 0 0 0 12 0 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
,
 12 0 0 0 0 0 12 1 0 0 0 0 0 0 8 12 0 0 0 0 11 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 8 0 0 0 0 3 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 2 8 0 0 0 0 0 0 12 1 0 0 0 0 0 1

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

C2×D6.6D6 in GAP, Magma, Sage, TeX

C_2\times D_6._6D_6
% in TeX

G:=Group("C2xD6.6D6");
// GroupNames label

G:=SmallGroup(288,949);
// by ID

G=gap.SmallGroup(288,949);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,346,80,1356,9414]);
// Polycyclic

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

׿
×
𝔽