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## G = C3×D4×A4order 288 = 25·32

### Direct product of C3, D4 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×D4×A4
 Chief series C1 — C22 — C23 — C22×C6 — C6×A4 — A4×C2×C6 — C3×D4×A4
 Lower central C22 — C23 — C3×D4×A4
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D4×A4
G = < a,b,c,d,e,f | a3=b4=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 516 in 164 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, D4, D4, C23, C23, C32, C12, C12, A4, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×C6, C2×C12, C3×D4, C3×D4, C2×A4, C2×A4, C22×C6, C22×C6, C22×D4, C3×C12, C3×A4, C62, C4×A4, C22×C12, C6×D4, C22×A4, C23×C6, D4×C32, C6×A4, C6×A4, D4×A4, D4×C2×C6, C12×A4, A4×C2×C6, C3×D4×A4
Quotients: C1, C2, C3, C22, C6, D4, C32, A4, C2×C6, C3×C6, C3×D4, C2×A4, C3×A4, C62, C22×A4, D4×C32, C6×A4, D4×A4, A4×C2×C6, C3×D4×A4

Smallest permutation representation of C3×D4×A4
On 36 points
Generators in S36
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 26 32)(6 27 29)(7 28 30)(8 25 31)(9 36 20)(10 33 17)(11 34 18)(12 35 19)
(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)
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 35 8)(2 36 5)(3 33 6)(4 34 7)(9 32 22)(10 29 23)(11 30 24)(12 31 21)(13 19 25)(14 20 26)(15 17 27)(16 18 28)

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

G:=Group( (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,26,32)(6,27,29)(7,28,30)(8,25,31)(9,36,20)(10,33,17)(11,34,18)(12,35,19), (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), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (5,7)(6,8)(9,11)(10,12)(17,19)(18,20)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,35,8)(2,36,5)(3,33,6)(4,34,7)(9,32,22)(10,29,23)(11,30,24)(12,31,21)(13,19,25)(14,20,26)(15,17,27)(16,18,28) );

G=PermutationGroup([[(1,13,21),(2,14,22),(3,15,23),(4,16,24),(5,26,32),(6,27,29),(7,28,30),(8,25,31),(9,36,20),(10,33,17),(11,34,18),(12,35,19)], [(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)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35)], [(5,7),(6,8),(9,11),(10,12),(17,19),(18,20),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36)], [(1,3),(2,4),(5,7),(6,8),(13,15),(14,16),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,35,8),(2,36,5),(3,33,6),(4,34,7),(9,32,22),(10,29,23),(11,30,24),(12,31,21),(13,19,25),(14,20,26),(15,17,27),(16,18,28)]])

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K ··· 6P 6Q 6R 6S 6T 6U ··· 6AF 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 4 4 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 2 2 3 3 6 6 1 1 4 ··· 4 2 6 1 1 2 2 2 2 3 3 3 3 4 ··· 4 6 6 6 6 8 ··· 8 2 2 6 6 8 ··· 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 6 6 type + + + + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 D4 C3×D4 C3×D4 A4 C2×A4 C2×A4 C3×A4 C6×A4 C6×A4 D4×A4 C3×D4×A4 kernel C3×D4×A4 C12×A4 A4×C2×C6 D4×A4 D4×C2×C6 C4×A4 C22×C12 C22×A4 C23×C6 C3×A4 A4 C2×C6 C3×D4 C12 C2×C6 D4 C4 C22 C3 C1 # reps 1 1 2 6 2 6 2 12 4 1 6 2 1 1 2 2 2 4 1 2

Matrix representation of C3×D4×A4 in GL5(𝔽13)

 3 0 0 0 0 0 3 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 0 12 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 0 12 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 12 12 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 12 12 12 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[0,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[0,12,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,12,0,1,0,0,12,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,12,0,0,0,1,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;

C3×D4×A4 in GAP, Magma, Sage, TeX

C_3\times D_4\times A_4
% in TeX

G:=Group("C3xD4xA4");
// GroupNames label

G:=SmallGroup(288,980);
// by ID

G=gap.SmallGroup(288,980);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,2,533,1531,2666]);
// Polycyclic

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

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