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## G = C3×D8⋊3S3order 288 = 25·32

### Direct product of C3 and D8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×D8⋊3S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×D4⋊2S3 — C3×D8⋊3S3
 Lower central C3 — C6 — C12 — C3×D8⋊3S3
 Upper central C1 — C6 — C12 — C3×D8

Generators and relations for C3×D83S3
G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 330 in 135 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3, C6 [×2], C6 [×8], C8, C8, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×4], D6, C2×C6 [×7], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3, C3×C6, C3×C6 [×2], C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×2], C4○D8, C3×Dic3, C3×Dic3 [×2], C3×C12, S3×C6, C62 [×2], S3×C8, Dic12, D4.S3 [×2], C2×C24, C3×D8 [×2], C3×D8, C3×SD16 [×2], C3×Q16, D42S3 [×2], C3×C4○D4 [×2], C3×C3⋊C8, C3×C24, C3×Dic6 [×2], S3×C12, C6×Dic3 [×2], C3×C3⋊D4 [×2], D4×C32 [×2], D83S3, C3×C4○D8, S3×C24, C3×Dic12, C3×D4.S3 [×2], C32×D8, C3×D42S3 [×2], C3×D83S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C4○D8, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, D83S3, C3×C4○D8, C3×S3×D4, C3×D83S3

Smallest permutation representation of C3×D83S3
On 48 points
Generators in S48
(1 25 40)(2 26 33)(3 27 34)(4 28 35)(5 29 36)(6 30 37)(7 31 38)(8 32 39)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(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 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 38)(34 37)(35 36)(39 40)(41 46)(42 45)(43 44)(47 48)
(1 40 25)(2 33 26)(3 34 27)(4 35 28)(5 36 29)(6 37 30)(7 38 31)(8 39 32)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L ··· 6Q 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 12M 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 12 12 12 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 4 4 6 1 1 2 2 2 2 3 3 12 12 1 1 2 2 2 4 4 4 4 6 6 8 ··· 8 2 2 6 6 2 2 3 3 3 3 4 4 4 12 12 12 12 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 C3×S3 C3×D4 C3×D4 C4○D8 S3×C6 S3×C6 C3×C4○D8 S3×D4 D8⋊3S3 C3×S3×D4 C3×D8⋊3S3 kernel C3×D8⋊3S3 S3×C24 C3×Dic12 C3×D4.S3 C32×D8 C3×D4⋊2S3 D8⋊3S3 S3×C8 Dic12 D4.S3 C3×D8 D4⋊2S3 C3×D8 C3×Dic3 S3×C6 C24 C3×D4 D8 Dic3 D6 C32 C8 D4 C3 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 1 1 2 2 2 2 4 2 4 8 1 2 2 4

Matrix representation of C3×D83S3 in GL4(𝔽7) generated by

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

C3×D83S3 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes_3S_3
% in TeX

G:=Group("C3xD8:3S3");
// GroupNames label

G:=SmallGroup(288,683);
// by ID

G=gap.SmallGroup(288,683);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,1094,303,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=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,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations

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