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## G = C2×D6.3D6order 288 = 25·32

### Direct product of C2 and D6.3D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×D6.3D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — C2×S3×Dic3 — C2×D6.3D6
 Lower central C32 — C3×C6 — C2×D6.3D6
 Upper central C1 — C22 — C23

Generators and relations for C2×D6.3D6
G = < a,b,c,d,e | a2=b6=c2=d6=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >

Subgroups: 1218 in 355 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×10], S3 [×8], C6 [×2], C6 [×4], C6 [×11], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C32, Dic3 [×6], Dic3 [×6], C12 [×6], D6 [×2], D6 [×14], C2×C6 [×2], C2×C6 [×4], C2×C6 [×13], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×8], C4×S3 [×12], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×7], C3⋊D4 [×4], C3⋊D4 [×16], C2×C12 [×7], C3×D4 [×4], C22×S3, C22×S3 [×3], C22×C6 [×2], C22×C6 [×2], C2×C4○D4, C3×Dic3 [×6], C3⋊Dic3 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C2×Dic6 [×2], S3×C2×C4 [×3], C2×D12, C4○D12 [×8], D42S3 [×8], C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4 [×4], C22×C12, C6×D4, S3×Dic3 [×4], C6.D6 [×4], C3⋊D12 [×4], C322Q8 [×4], C6×Dic3 [×3], C6×Dic3 [×4], C3×C3⋊D4 [×4], C2×C3⋊Dic3, C327D4 [×4], S3×C2×C6, C22×C3⋊S3, C2×C62, C2×C4○D12, C2×D42S3, C2×S3×Dic3, D6.3D6 [×8], C2×C6.D6, C2×C3⋊D12, C2×C322Q8, Dic3×C2×C6, C6×C3⋊D4, C2×C327D4, C2×D6.3D6
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], D42S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×C4○D12, C2×D42S3, D6.3D6 [×2], C22×S32, C2×D6.3D6

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

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

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

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

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 ··· 6J 6K ··· 6S 6T 6U 12A ··· 12H 12I 12J 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 12 ··· 12 12 12 size 1 1 1 1 2 2 6 6 18 18 2 2 4 3 3 3 3 6 6 6 6 18 18 2 ··· 2 4 ··· 4 12 12 6 ··· 6 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 C4○D4 C4○D12 S32 D4⋊2S3 C2×S32 D6.3D6 kernel C2×D6.3D6 C2×S3×Dic3 D6.3D6 C2×C6.D6 C2×C3⋊D12 C2×C32⋊2Q8 Dic3×C2×C6 C6×C3⋊D4 C2×C32⋊7D4 C22×Dic3 C2×C3⋊D4 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C3×C6 C6 C23 C6 C22 C2 # reps 1 1 8 1 1 1 1 1 1 1 1 7 4 1 2 4 8 1 2 3 4

Matrix representation of C2×D6.3D6 in GL6(𝔽13)

 1 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 12 0 0 0 0 0 0 12
,
 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
,
 1 0 0 0 0 0 1 12 0 0 0 0 0 0 0 5 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 0 12 1

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

C2×D6.3D6 in GAP, Magma, Sage, TeX

C_2\times D_6._3D_6
% in TeX

G:=Group("C2xD6.3D6");
// GroupNames label

G:=SmallGroup(288,970);
// by ID

G=gap.SmallGroup(288,970);
# by ID

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

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

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