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## G = D4×C3.A4order 288 = 25·32

### Direct product of D4 and C3.A4

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C3.A4
G = < a,b,c,d,e,f | a4=b2=c3=d2=e2=1, f3=c, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 366 in 116 conjugacy classes, 30 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, D4, D4, C23, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C18, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, C22×D4, C36, C3.A4, C2×C18, C22×C12, C6×D4, C23×C6, D4×C9, C2×C3.A4, C2×C3.A4, D4×C2×C6, C4×C3.A4, C22×C3.A4, D4×C3.A4
Quotients: C1, C2, C3, C22, C6, D4, C9, A4, C2×C6, C18, C3×D4, C2×A4, C3.A4, C2×C18, C22×A4, D4×C9, C2×C3.A4, D4×A4, C22×C3.A4, D4×C3.A4

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 18G ··· 18R 36A ··· 36F order 1 2 2 2 2 2 2 2 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 3 3 6 6 1 1 2 6 1 1 2 2 2 2 3 3 3 3 6 6 6 6 4 ··· 4 2 2 6 6 4 ··· 4 8 ··· 8 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 C6 C6 C9 C18 C18 D4 C3×D4 D4×C9 A4 C2×A4 C2×A4 C3.A4 C2×C3.A4 C2×C3.A4 D4×A4 D4×C3.A4 kernel D4×C3.A4 C4×C3.A4 C22×C3.A4 D4×C2×C6 C22×C12 C23×C6 C22×D4 C22×C4 C24 C3.A4 C2×C6 C22 C3×D4 C12 C2×C6 D4 C4 C22 C3 C1 # reps 1 1 2 2 2 4 6 6 12 1 2 6 1 1 2 2 2 4 1 2

Matrix representation of D4×C3.A4 in GL5(𝔽37)

 0 1 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 36 36 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 36 36 36 0 0 1 0 0
,
 26 0 0 0 0 0 26 0 0 0 0 0 0 0 33 0 0 33 0 0 0 0 0 33 0

G:=sub<GL(5,GF(37))| [0,36,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[26,0,0,0,0,0,26,0,0,0,0,0,0,33,0,0,0,0,0,33,0,0,33,0,0] >;

D4×C3.A4 in GAP, Magma, Sage, TeX

D_4\times C_3.A_4
% in TeX

G:=Group("D4xC3.A4");
// GroupNames label

G:=SmallGroup(288,344);
// by ID

G=gap.SmallGroup(288,344);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,142,1531,2666]);
// Polycyclic

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