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## G = C3×Q8.14D6order 288 = 25·32

### Direct product of C3 and Q8.14D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Q8.14D6
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×Dic6 — C6×Dic6 — C3×Q8.14D6
 Lower central C3 — C6 — C12 — C3×Q8.14D6
 Upper central C1 — C6 — C2×C12 — C3×C4○D4

Generators and relations for C3×Q8.14D6
G = < a,b,c,d,e | a3=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >

Subgroups: 290 in 140 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6 [×2], C6 [×8], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C32, Dic3 [×2], C12 [×4], C12 [×8], C2×C6 [×2], C2×C6 [×5], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8 [×4], C8.C22, C3×Dic3 [×2], C3×C12 [×2], C3×C12, C62, C62, C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C2×Dic6, C6×Q8, C3×C4○D4 [×2], C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6 [×2], C3×Dic6, C6×Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.14D6, C3×C8.C22, C3×C4.Dic3, C3×D4.S3 [×2], C3×C3⋊Q16 [×2], C6×Dic6, C32×C4○D4, C3×Q8.14D6
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], C3×D4 [×2], C22×S3, C22×C6, C8.C22, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, Q8.14D6, C3×C8.C22, C6×C3⋊D4, C3×Q8.14D6

Smallest permutation representation of C3×Q8.14D6
On 48 points
Generators in S48
(1 3 2)(4 5 6)(7 9 8)(10 12 11)(13 15 14)(16 17 18)(19 20 21)(22 23 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 11 15 8)(2 12 13 9)(3 10 14 7)(4 22 21 16)(5 23 19 17)(6 24 20 18)(25 35 28 32)(26 36 29 33)(27 31 30 34)(37 46 40 43)(38 47 41 44)(39 48 42 45)
(1 36 15 33)(2 34 13 31)(3 32 14 35)(4 43 21 46)(5 47 19 44)(6 45 20 48)(7 25 10 28)(8 29 11 26)(9 27 12 30)(16 37 22 40)(17 41 23 38)(18 39 24 42)
(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 20 15 6)(2 19 13 5)(3 21 14 4)(7 16 10 22)(8 18 11 24)(9 17 12 23)(25 43 28 46)(26 48 29 45)(27 47 30 44)(31 38 34 41)(32 37 35 40)(33 42 36 39)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6G 6H ··· 6R 8A 8B 12A ··· 12J 12K ··· 12U 12V 12W 12X 12Y 24A 24B 24C 24D order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 12 12 12 12 24 24 24 24 size 1 1 2 4 1 1 2 2 2 2 2 4 12 12 1 1 2 ··· 2 4 ··· 4 12 12 2 ··· 2 4 ··· 4 12 12 12 12 12 12 12 12

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 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 D6 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 S3×C6 C3×C3⋊D4 C3×C3⋊D4 C8.C22 Q8.14D6 C3×C8.C22 C3×Q8.14D6 kernel C3×Q8.14D6 C3×C4.Dic3 C3×D4.S3 C3×C3⋊Q16 C6×Dic6 C32×C4○D4 Q8.14D6 C4.Dic3 D4.S3 C3⋊Q16 C2×Dic6 C3×C4○D4 C3×C4○D4 C3×C12 C62 C2×C12 C3×D4 C3×Q8 C4○D4 C12 C12 C2×C6 C2×C6 C2×C4 D4 Q8 C4 C22 C32 C3 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 1 2 2 4

Matrix representation of C3×Q8.14D6 in GL6(𝔽73)

 8 0 0 0 0 0 0 8 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 48 0 0 0 0 3 72 0 0 0 0 0 60 72 2 0 0 56 60 72 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 56 0 70 0 0 1 0 0 0 0 0 0 48 0 17
,
 9 0 0 0 0 0 64 65 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 13 0 72
,
 65 66 0 0 0 0 9 8 0 0 0 0 0 0 0 4 0 0 0 0 18 0 0 0 0 0 0 68 61 12 0 0 44 68 67 12

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,56,0,0,48,72,60,60,0,0,0,0,72,72,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,56,0,48,0,0,72,0,0,0,0,0,0,70,0,17],[9,64,0,0,0,0,0,65,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,13,0,0,0,0,72,0,0,0,0,0,0,72],[65,9,0,0,0,0,66,8,0,0,0,0,0,0,0,18,0,44,0,0,4,0,68,68,0,0,0,0,61,67,0,0,0,0,12,12] >;

C3×Q8.14D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{14}D_6
% in TeX

G:=Group("C3xQ8.14D6");
// GroupNames label

G:=SmallGroup(288,722);
// by ID

G=gap.SmallGroup(288,722);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,555,2524,648,102,9414]);
// 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=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

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