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## G = C6×C4.Dic3order 288 = 25·32

### Direct product of C6 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×C3⋊C8 — C6×C3⋊C8 — C6×C4.Dic3
 Lower central C3 — C6 — C6×C4.Dic3
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C6×C4.Dic3
G = < a,b,c,d | a6=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 250 in 163 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C22×C6, C22×C6, C2×M4(2), C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C22×C12, C22×C12, C3×C3⋊C8, C6×C12, C6×C12, C2×C62, C2×C4.Dic3, C6×M4(2), C6×C3⋊C8, C3×C4.Dic3, C2×C6×C12, C6×C4.Dic3
Quotients:

Smallest permutation representation of C6×C4.Dic3
On 48 points
Generators in S48
(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 45 29 37 33 41)(26 46 30 38 34 42)(27 47 31 39 35 43)(28 48 32 40 36 44)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 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 32 10 29 7 26 4 35)(2 25 11 34 8 31 5 28)(3 30 12 27 9 36 6 33)(13 44 22 41 19 38 16 47)(14 37 23 46 20 43 17 40)(15 42 24 39 21 48 18 45)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6AE 8A ··· 8H 12A ··· 12H 12I ··· 12AJ 24A ··· 24P order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 1 1 2 2 2 1 1 1 1 2 2 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

108 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 type + + + + + - + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×Dic3 S3×C6 C3×Dic3 C4.Dic3 C3×M4(2) C3×C4.Dic3 kernel C6×C4.Dic3 C6×C3⋊C8 C3×C4.Dic3 C2×C6×C12 C2×C4.Dic3 C6×C12 C2×C62 C2×C3⋊C8 C4.Dic3 C22×C12 C2×C12 C22×C6 C22×C12 C2×C12 C2×C12 C22×C6 C3×C6 C22×C4 C2×C4 C2×C4 C23 C6 C6 C2 # reps 1 2 4 1 2 6 2 4 8 2 12 4 1 3 3 1 4 2 6 6 2 8 8 16

Matrix representation of C6×C4.Dic3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 65 0 0 0 0 65
,
 27 0 0 0 0 46 0 0 0 0 1 0 0 0 0 1
,
 27 0 0 0 0 27 0 0 0 0 8 0 0 0 0 64
,
 0 1 0 0 27 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,65,0,0,0,0,65],[27,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,8,0,0,0,0,64],[0,27,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C6×C4.Dic3 in GAP, Magma, Sage, TeX

C_6\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C6xC4.Dic3");
// GroupNames label

G:=SmallGroup(288,692);
// by ID

G=gap.SmallGroup(288,692);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,1094,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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