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## G = C2×D6.D6order 288 = 25·32

### Direct product of C2 and D6.D6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×D6.D6
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=d5 >

Subgroups: 1234 in 355 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×12], C6 [×6], C6 [×7], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×6], D6 [×4], D6 [×18], C2×C6 [×2], C2×C6 [×9], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×8], C4×S3 [×8], C4×S3 [×12], D12 [×8], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×16], C2×C12 [×2], C2×C12 [×11], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C2×C4○D4, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C2×Dic6 [×2], S3×C2×C4 [×2], S3×C2×C4 [×3], C2×D12 [×2], C4○D12 [×16], C2×C3⋊D4 [×4], C22×C12 [×2], D6⋊S3 [×4], C3⋊D12 [×8], C322Q8 [×4], S3×C12 [×8], C6×Dic3 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, C2×C4○D12 [×2], D6.D6 [×8], C2×D6⋊S3, C2×C3⋊D12 [×2], C2×C322Q8, S3×C2×C12 [×2], C2×C4×C3⋊S3, C2×D6.D6
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 [×4], S3×C23 [×2], C2×S32 [×3], C2×C4○D12 [×2], D6.D6 [×2], C22×S32, C2×D6.D6

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

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

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

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

60 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 ··· 6Q 12A ··· 12H 12I 12J 12K 12L 12M ··· 12T 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 12 ··· 12 12 12 12 12 12 ··· 12 size 1 1 1 1 6 6 6 6 18 18 2 2 4 1 1 1 1 6 6 6 6 18 18 2 ··· 2 4 4 4 6 ··· 6 2 ··· 2 4 4 4 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 C4○D4 C4○D12 S32 C2×S32 C2×S32 D6.D6 kernel C2×D6.D6 D6.D6 C2×D6⋊S3 C2×C3⋊D12 C2×C32⋊2Q8 S3×C2×C12 C2×C4×C3⋊S3 S3×C2×C4 C4×S3 C2×Dic3 C2×C12 C22×S3 C3×C6 C6 C2×C4 C4 C22 C2 # reps 1 8 1 2 1 2 1 2 8 2 2 2 4 16 1 2 1 4

Matrix representation of C2×D6.D6 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 12 0 0 0 0 0 0 12
,
 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 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 12
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12

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

C2×D6.D6 in GAP, Magma, Sage, TeX

C_2\times D_6.D_6
% in TeX

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

G:=SmallGroup(288,948);
// by ID

G=gap.SmallGroup(288,948);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,80,1356,9414]);
// 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=d^5>;
// generators/relations

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