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## G = C3×C23⋊2D6order 288 = 25·32

### Direct product of C3 and C23⋊2D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23⋊2D6
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C22×C6 — C3×C23⋊2D6
 Lower central C3 — C2×C6 — C3×C23⋊2D6
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C232D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 778 in 287 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×4], C6 [×2], C6 [×4], C6 [×17], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], C32, Dic3 [×2], C12 [×6], D6 [×4], D6 [×12], C2×C6 [×2], C2×C6 [×4], C2×C6 [×35], C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, C3×S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×12], C22×S3 [×2], C22×S3 [×6], C22×C6 [×4], C22×C6 [×10], C22≀C2, C3×Dic3 [×2], C3×C12, S3×C6 [×4], S3×C6 [×12], C62, C62 [×2], C62 [×5], D6⋊C4 [×2], C6.D4, C3×C22⋊C4 [×3], C2×C3⋊D4 [×2], C6×D4 [×2], C6×D4 [×3], S3×C23, C23×C6, C6×Dic3 [×2], C3×C3⋊D4 [×4], C6×C12, D4×C32 [×2], S3×C2×C6 [×2], S3×C2×C6 [×6], C2×C62 [×2], C232D6, C3×C22≀C2, C3×D6⋊C4 [×2], C3×C6.D4, C6×C3⋊D4 [×2], D4×C3×C6, S3×C22×C6, C3×C232D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×6], C23, D6 [×3], C2×C6 [×7], C2×D4 [×3], C3×S3, C3⋊D4 [×2], C3×D4 [×6], C22×S3, C22×C6, C22≀C2, S3×C6 [×3], S3×D4 [×2], C2×C3⋊D4, C6×D4 [×3], C3×C3⋊D4 [×2], S3×C2×C6, C232D6, C3×C22≀C2, C3×S3×D4 [×2], C6×C3⋊D4, C3×C232D6

Smallest permutation representation of C3×C232D6
On 48 points
Generators in S48
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 27)(2 21)(3 29)(4 23)(5 25)(6 19)(7 38)(8 31)(9 40)(10 33)(11 42)(12 35)(13 20)(14 28)(15 22)(16 30)(17 24)(18 26)(32 45)(34 47)(36 43)(37 48)(39 44)(41 46)
(1 4)(2 5)(3 6)(7 46)(8 47)(9 48)(10 43)(11 44)(12 45)(13 16)(14 17)(15 18)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 36)(7 29)(8 28)(9 27)(10 26)(11 25)(12 30)(13 37)(14 42)(15 41)(16 40)(17 39)(18 38)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3A 3B 3C 3D 3E 4A 4B 4C 6A ··· 6F 6G ··· 6S 6T ··· 6AG 6AH ··· 6AO 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 4 6 6 6 6 1 1 2 2 2 4 12 12 1 ··· 1 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4 12 12 12 12

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×C3⋊D4 S3×D4 C3×S3×D4 kernel C3×C23⋊2D6 C3×D6⋊C4 C3×C6.D4 C6×C3⋊D4 D4×C3×C6 S3×C22×C6 C23⋊2D6 D6⋊C4 C6.D4 C2×C3⋊D4 C6×D4 S3×C23 C6×D4 S3×C6 C62 C2×C12 C22×C6 C2×D4 D6 C2×C6 C2×C6 C2×C4 C23 C22 C6 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 1 4 2 1 2 2 8 4 4 2 4 8 2 4

Matrix representation of C3×C232D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 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 12 5 0 0 0 0 0 1
,
 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 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 12 0 0 0 0 0 0 9 0 0 0 0 0 0 3 0 0 0 0 0 0 12 0 0 0 0 0 10 1
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 3 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C3×C232D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes_2D_6
% in TeX

G:=Group("C3xC2^3:2D6");
// GroupNames label

G:=SmallGroup(288,708);
// by ID

G=gap.SmallGroup(288,708);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,9414]);
// Polycyclic

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

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