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## G = C3×Q8⋊3D6order 288 = 25·32

### Direct product of C3 and Q8⋊3D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Q8⋊3D6
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×D4 — C3×Q8⋊3D6
 Lower central C3 — C6 — C12 — C3×Q8⋊3D6
 Upper central C1 — C6 — C12 — C3×SD16

Generators and relations for C3×Q83D6
G = < a,b,c,d,e | a3=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 434 in 146 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×3], C6 [×2], C6 [×7], C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3, C12 [×2], C12 [×5], D6, D6 [×4], C2×C6 [×8], M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C3×S3 [×3], C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C4×S3, D12 [×2], D12, C3⋊D4, C2×C12 [×2], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6 [×4], C62, C8⋊S3, D24, D4⋊S3, Q82S3, C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C3×SD16 [×2], S3×D4, Q83S3, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×C24, S3×C12, S3×C12, C3×D12 [×2], C3×D12, C3×C3⋊D4, D4×C32, Q8×C32, S3×C2×C6, Q83D6, C3×C8⋊C22, C3×C8⋊S3, C3×D24, C3×D4⋊S3, C3×Q82S3, C32×SD16, C3×S3×D4, C3×Q83S3, C3×Q83D6
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, C8⋊C22, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, Q83D6, C3×C8⋊C22, C3×S3×D4, C3×Q83D6

Smallest permutation representation of C3×Q83D6
On 48 points
Generators in S48
(1 2 3)(4 6 5)(7 9 8)(10 11 12)(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 31 12 34)(2 35 10 32)(3 33 11 36)(4 25 9 28)(5 29 7 26)(6 27 8 30)(13 16 20 23)(14 24 21 17)(15 18 22 19)(37 43 46 40)(38 41 47 44)(39 45 48 42)
(1 43 12 40)(2 47 10 38)(3 45 11 42)(4 17 9 24)(5 15 7 22)(6 13 8 20)(14 25 21 28)(16 27 23 30)(18 29 19 26)(31 37 34 46)(32 44 35 41)(33 39 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 5)(2 4)(3 6)(7 12)(8 11)(9 10)(13 39)(14 38)(15 37)(16 42)(17 41)(18 40)(19 43)(20 48)(21 47)(22 46)(23 45)(24 44)(25 32)(26 31)(27 36)(28 35)(29 34)(30 33)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 8A 8B 12A 12B 12C ··· 12G 12H 12I 12J 12K 12L 24A ··· 24H 24I 24J order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 8 8 12 12 12 ··· 12 12 12 12 12 12 24 ··· 24 24 24 size 1 1 4 6 12 12 1 1 2 2 2 2 4 6 1 1 2 2 2 4 4 6 6 8 8 8 12 12 12 12 4 12 2 2 4 ··· 4 6 6 8 8 8 4 ··· 4 12 12

54 irreducible representations

 dim 1 1 1 1 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 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 D6 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 S3×C6 C8⋊C22 S3×D4 Q8⋊3D6 C3×C8⋊C22 C3×S3×D4 C3×Q8⋊3D6 kernel C3×Q8⋊3D6 C3×C8⋊S3 C3×D24 C3×D4⋊S3 C3×Q8⋊2S3 C32×SD16 C3×S3×D4 C3×Q8⋊3S3 Q8⋊3D6 C8⋊S3 D24 D4⋊S3 Q8⋊2S3 C3×SD16 S3×D4 Q8⋊3S3 C3×SD16 C3×Dic3 S3×C6 C24 C3×D4 C3×Q8 SD16 Dic3 D6 C8 D4 Q8 C32 C6 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 1 1 2 2 2 4

Matrix representation of C3×Q83D6 in GL8(𝔽73)

 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 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
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 71 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 72 2 0 0 0 0 0 0 72 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0
,
 8 0 0 0 0 0 0 0 0 65 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 71 0 0 0 0 0 0 0 72
,
 0 0 64 0 0 0 0 0 0 0 0 9 0 0 0 0 8 0 0 0 0 0 0 0 0 65 0 0 0 0 0 0 0 0 0 0 37 0 44 29 0 0 0 0 37 36 22 0 0 0 0 0 0 44 36 1 0 0 0 0 51 44 0 37

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,71,72],[0,0,8,0,0,0,0,0,0,0,0,65,0,0,0,0,64,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,37,37,0,51,0,0,0,0,0,36,44,44,0,0,0,0,44,22,36,0,0,0,0,0,29,0,1,37] >;

C3×Q83D6 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_3D_6
% in TeX

G:=Group("C3xQ8:3D6");
// GroupNames label

G:=SmallGroup(288,685);
// by ID

G=gap.SmallGroup(288,685);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,303,268,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=d^6=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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