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## G = C3×Q8.7D6order 288 = 25·32

### Direct product of C3 and Q8.7D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Q8.7D6
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×D4⋊2S3 — C3×Q8.7D6
 Lower central C3 — C6 — C12 — C3×Q8.7D6
 Upper central C1 — C6 — C12 — C3×SD16

Generators and relations for C3×Q8.7D6
G = < a,b,c,d,e | a3=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >

Subgroups: 338 in 134 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×2], C6 [×2], C6 [×6], C8, C8, C2×C4 [×3], D4, D4 [×3], Q8, Q8, C32, Dic3, Dic3, C12 [×2], C12 [×6], D6, D6, C2×C6 [×5], C2×C8, D8, SD16, SD16, Q16, C4○D4 [×2], C3×S3 [×2], C3×C6, C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12 [×3], C3×D4 [×2], C3×D4 [×4], C3×Q8 [×2], C3×Q8 [×2], C4○D8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16 [×2], C3×SD16 [×2], C3×Q16, D42S3, Q83S3, C3×C4○D4 [×2], C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C6×Dic3, C3×C3⋊D4, D4×C32, Q8×C32, Q8.7D6, C3×C4○D8, S3×C24, C3×C24⋊C2, C3×D4⋊S3, C3×C3⋊Q16, C32×SD16, C3×D42S3, C3×Q83S3, C3×Q8.7D6
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, Q8.7D6, C3×C4○D8, C3×S3×D4, C3×Q8.7D6

Smallest permutation representation of C3×Q8.7D6
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 23 21)(20 24 22)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 19 10 22)(2 23 11 20)(3 21 12 24)(4 13 8 16)(5 17 9 14)(6 15 7 18)(25 28 42 39)(26 40 37 29)(27 30 38 41)(31 45 48 34)(32 35 43 46)(33 47 44 36)
(1 43 10 32)(2 47 11 36)(3 45 12 34)(4 27 8 38)(5 25 9 42)(6 29 7 40)(13 41 16 30)(14 28 17 39)(15 37 18 26)(19 35 22 46)(20 44 23 33)(21 31 24 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 14 10 17)(2 16 11 13)(3 18 12 15)(4 23 8 20)(5 19 9 22)(6 21 7 24)(25 32 42 43)(26 48 37 31)(27 36 38 47)(28 46 39 35)(29 34 40 45)(30 44 41 33)

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

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

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

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 6M 6N 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12K 12L 12M 12N 12O 12P 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 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 6 12 1 1 2 2 2 2 3 3 4 12 1 1 2 2 2 4 4 6 6 8 8 8 12 12 2 2 6 6 2 2 3 3 3 3 4 ··· 4 8 8 8 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 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 C4○D8 S3×C6 S3×C6 S3×C6 C3×C4○D8 S3×D4 Q8.7D6 C3×S3×D4 C3×Q8.7D6 kernel C3×Q8.7D6 S3×C24 C3×C24⋊C2 C3×D4⋊S3 C3×C3⋊Q16 C32×SD16 C3×D4⋊2S3 C3×Q8⋊3S3 Q8.7D6 S3×C8 C24⋊C2 D4⋊S3 C3⋊Q16 C3×SD16 D4⋊2S3 Q8⋊3S3 C3×SD16 C3×Dic3 S3×C6 C24 C3×D4 C3×Q8 SD16 Dic3 D6 C32 C8 D4 Q8 C3 C6 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 4 2 2 2 8 1 2 2 4

Matrix representation of C3×Q8.7D6 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 1 71 0 0 1 72 0 0 0 0 1 0 0 0 0 1
,
 61 12 0 0 67 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 1 72 0 0 0 0 64 0 0 0 0 8
,
 27 19 0 0 0 46 0 0 0 0 0 8 0 0 64 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,71,72,0,0,0,0,1,0,0,0,0,1],[61,67,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,72,0,0,0,0,64,0,0,0,0,8],[27,0,0,0,19,46,0,0,0,0,0,64,0,0,8,0] >;

C3×Q8.7D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._7D_6
% in TeX

G:=Group("C3xQ8.7D6");
// GroupNames label

G:=SmallGroup(288,687);
// by ID

G=gap.SmallGroup(288,687);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,1094,303,268,1271,648,102,9414]);
// Polycyclic

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

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