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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 — C2×S32 — C22×S32 — C2×D6⋊D6
 Lower central C32 — C3×C6 — C2×D6⋊D6
 Upper central C1 — C22 — C2×C4

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

Subgroups: 2130 in 499 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×38], S3 [×20], C6 [×6], C6 [×11], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], C32, Dic3 [×6], C12 [×4], C12 [×2], D6 [×8], D6 [×58], C2×C6 [×2], C2×C6 [×17], C22×C4, C2×D4 [×12], C24 [×2], C3×S3 [×8], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×12], D12 [×8], C2×Dic3 [×3], C3⋊D4 [×16], C2×C12 [×2], C2×C12, C3×D4 [×8], C22×S3 [×4], C22×S3 [×35], C22×C6 [×4], C22×D4, C3⋊Dic3 [×2], C3×C12 [×2], S32 [×16], S3×C6 [×8], S3×C6 [×8], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×3], C2×D12 [×2], S3×D4 [×16], C2×C3⋊D4 [×4], C6×D4 [×2], S3×C23 [×4], D6⋊S3 [×8], C3×D12 [×8], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C2×S32 [×8], C2×S32 [×8], S3×C2×C6 [×4], C22×C3⋊S3, C2×S3×D4 [×2], D6⋊D6 [×8], C2×D6⋊S3 [×2], C6×D12 [×2], C2×C4×C3⋊S3, C22×S32 [×2], C2×D6⋊D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], D4 [×4], C23 [×15], D6 [×14], C2×D4 [×6], C24, C22×S3 [×14], C22×D4, S32, S3×D4 [×4], S3×C23 [×2], C2×S32 [×3], C2×S3×D4 [×2], D6⋊D6 [×2], C22×S32, C2×D6⋊D6

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 3A 3B 3C 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J ··· 6Q 12A ··· 12H order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 ··· 6 12 ··· 12 size 1 1 1 1 6 ··· 6 9 9 9 9 2 2 4 2 2 18 18 2 ··· 2 4 4 4 12 ··· 12 4 ··· 4

48 irreducible representations

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

Matrix representation of C2×D6⋊D6 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 8 0 0 0 0 0 0 11 11 0 0 0 0 0 0 0 0 7 11 0 0 0 0 0 0 11 6 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 12
,
 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 7 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 0 4 10 0 0 0 0 0 0 5 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 10 6 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 9 3 0 0 0 0 0 0 8 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,11,0,0,0,0,0,0,8,11,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,11,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12],[1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,3,10,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,4,5,0,0,0,0,0,0,10,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,3,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,9,8,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C2×D6⋊D6 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes D_6
% in TeX

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

G:=SmallGroup(288,952);
// by ID

G=gap.SmallGroup(288,952);
# by ID

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

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

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