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## G = C3×S3×D8order 288 = 25·32

### Direct product of C3, S3 and D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×S3×D8
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×D4 — C3×S3×D8
 Lower central C3 — C6 — C12 — C3×S3×D8
 Upper central C1 — C6 — C12 — C3×D8

Generators and relations for C3×S3×D8
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 522 in 163 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×2], S3 [×2], C6 [×2], C6 [×11], C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], C32, Dic3, C12 [×2], C12 [×2], D6, D6 [×6], C2×C6 [×13], C2×C8, D8, D8 [×3], C2×D4 [×2], C3×S3 [×2], C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8, C24 [×2], C24 [×2], C4×S3, D12 [×2], C3⋊D4 [×2], C2×C12, C3×D4 [×4], C3×D4 [×6], C22×S3 [×2], C22×C6 [×2], C2×D8, C3×Dic3, C3×C12, S3×C6, S3×C6 [×6], C62 [×2], S3×C8, D24, D4⋊S3 [×2], C2×C24, C3×D8 [×2], C3×D8 [×4], S3×D4 [×2], C6×D4 [×2], C3×C3⋊C8, C3×C24, S3×C12, C3×D12 [×2], C3×C3⋊D4 [×2], D4×C32 [×2], S3×C2×C6 [×2], S3×D8, C6×D8, S3×C24, C3×D24, C3×D4⋊S3 [×2], C32×D8, C3×S3×D4 [×2], C3×S3×D8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], D8 [×2], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C2×D8, S3×C6 [×3], C3×D8 [×2], S3×D4, C6×D4, S3×C2×C6, S3×D8, C6×D8, C3×S3×D4, C3×S3×D8

Smallest permutation representation of C3×S3×D8
On 48 points
Generators in S48
(1 37 27)(2 38 28)(3 39 29)(4 40 30)(5 33 31)(6 34 32)(7 35 25)(8 36 26)(9 24 45)(10 17 46)(11 18 47)(12 19 48)(13 20 41)(14 21 42)(15 22 43)(16 23 44)
(1 37 27)(2 38 28)(3 39 29)(4 40 30)(5 33 31)(6 34 32)(7 35 25)(8 36 26)(9 45 24)(10 46 17)(11 47 18)(12 48 19)(13 41 20)(14 42 21)(15 43 22)(16 44 23)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 29)(26 28)(30 32)(34 40)(35 39)(36 38)(41 43)(44 48)(45 47)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N ··· 6S 6T 6U 6V 6W 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 3 3 4 4 12 12 1 1 2 2 2 2 6 1 1 2 2 2 3 3 3 3 4 4 4 4 8 ··· 8 12 12 12 12 2 2 6 6 2 2 4 4 4 6 6 2 2 2 2 4 ··· 4 6 6 6 6

63 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 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 D8 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 C3×D8 S3×D4 S3×D8 C3×S3×D4 C3×S3×D8 kernel C3×S3×D8 S3×C24 C3×D24 C3×D4⋊S3 C32×D8 C3×S3×D4 S3×D8 S3×C8 D24 D4⋊S3 C3×D8 S3×D4 C3×D8 C3×Dic3 S3×C6 C24 C3×D4 C3×S3 D8 Dic3 D6 C8 D4 S3 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 1 1 2 4 2 2 2 2 4 8 1 2 2 4

Matrix representation of C3×S3×D8 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 2 3 3 1 6 1 1 4 2 2 0 6 4 3 2 2
,
 4 2 5 3 6 2 1 4 0 6 5 4 5 0 3 3
,
 3 3 0 2 1 3 1 1 2 2 6 3 3 4 2 3
,
 2 0 6 1 6 4 6 6 2 4 1 2 6 4 3 0
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,6,2,4,3,1,2,3,3,1,0,2,1,4,6,2],[4,6,0,5,2,2,6,0,5,1,5,3,3,4,4,3],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[2,6,2,6,0,4,4,4,6,6,1,3,1,6,2,0] >;

C3×S3×D8 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_8
% in TeX

G:=Group("C3xS3xD8");
// GroupNames label

G:=SmallGroup(288,681);
// by ID

G=gap.SmallGroup(288,681);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,303,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=1,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,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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