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## G = C2×D4×A4order 192 = 26·3

### Direct product of C2, D4 and A4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×D4×A4
G = < a,b,c,d,e,f | a2=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: 1112 in 317 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22 [×2], C22 [×4], C22 [×55], C6 [×7], C2×C4, C2×C4 [×9], D4 [×4], D4 [×20], C23, C23 [×4], C23 [×62], C12 [×2], A4, C2×C6 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×37], C24, C24 [×4], C24 [×16], C2×C12, C3×D4 [×4], C2×A4, C2×A4 [×2], C2×A4 [×4], C22×C6 [×2], C23×C4, C22×D4 [×4], C22×D4 [×8], C25 [×2], C4×A4 [×2], C6×D4, C22×A4, C22×A4 [×4], C22×A4 [×4], D4×C23, C2×C4×A4, D4×A4 [×4], C23×A4 [×2], C2×D4×A4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, A4, C2×C6 [×7], C2×D4, C3×D4 [×2], C2×A4 [×7], C22×C6, C6×D4, C22×A4 [×7], D4×A4 [×2], C23×A4, C2×D4×A4

Permutation representations of C2×D4×A4
On 24 points - transitive group 24T411
Generators in S24
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)
(1 3)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 23)(18 24)(19 21)(20 22)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)

G:=sub<Sym(24)| (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24)>;

G:=Group( (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24) );

G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,20),(18,19),(21,24),(22,23)], [(1,3),(2,4),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12),(17,21),(18,22),(19,23),(20,24)], [(1,11),(2,12),(3,9),(4,10),(5,7),(6,8),(13,15),(14,16),(17,23),(18,24),(19,21),(20,22)], [(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,16,21),(10,13,22),(11,14,23),(12,15,24)])

G:=TransitiveGroup(24,411);

On 24 points - transitive group 24T412
Generators in S24
(1 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)(17 21)(18 22)(19 23)(20 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)(17 24)(18 23)(19 22)(20 21)
(9 14)(10 15)(11 16)(12 13)(17 21)(18 22)(19 23)(20 24)
(1 6)(2 7)(3 8)(4 5)(17 21)(18 22)(19 23)(20 24)
(1 15 23)(2 16 24)(3 13 21)(4 14 22)(5 9 18)(6 10 19)(7 11 20)(8 12 17)

G:=sub<Sym(24)| (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)(17,24)(18,23)(19,22)(20,21), (9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24), (1,6)(2,7)(3,8)(4,5)(17,21)(18,22)(19,23)(20,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)>;

G:=Group( (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)(17,24)(18,23)(19,22)(20,21), (9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24), (1,6)(2,7)(3,8)(4,5)(17,21)(18,22)(19,23)(20,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17) );

G=PermutationGroup([(1,6),(2,7),(3,8),(4,5),(9,14),(10,15),(11,16),(12,13),(17,21),(18,22),(19,23),(20,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,5),(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(12,16),(17,24),(18,23),(19,22),(20,21)], [(9,14),(10,15),(11,16),(12,13),(17,21),(18,22),(19,23),(20,24)], [(1,6),(2,7),(3,8),(4,5),(17,21),(18,22),(19,23),(20,24)], [(1,15,23),(2,16,24),(3,13,21),(4,14,22),(5,9,18),(6,10,19),(7,11,20),(8,12,17)])

G:=TransitiveGroup(24,412);

On 24 points - transitive group 24T413
Generators in S24
(1 21)(2 22)(3 23)(4 24)(5 15)(6 16)(7 13)(8 14)(9 17)(10 18)(11 19)(12 20)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(10 12)(13 15)(18 20)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 14 11)(2 15 12)(3 16 9)(4 13 10)(5 20 22)(6 17 23)(7 18 24)(8 19 21)

G:=sub<Sym(24)| (1,21)(2,22)(3,23)(4,24)(5,15)(6,16)(7,13)(8,14)(9,17)(10,18)(11,19)(12,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,14,11)(2,15,12)(3,16,9)(4,13,10)(5,20,22)(6,17,23)(7,18,24)(8,19,21)>;

G:=Group( (1,21)(2,22)(3,23)(4,24)(5,15)(6,16)(7,13)(8,14)(9,17)(10,18)(11,19)(12,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,14,11)(2,15,12)(3,16,9)(4,13,10)(5,20,22)(6,17,23)(7,18,24)(8,19,21) );

G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,15),(6,16),(7,13),(8,14),(9,17),(10,18),(11,19),(12,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(2,4),(5,7),(10,12),(13,15),(18,20),(22,24)], [(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24)], [(1,14,11),(2,15,12),(3,16,9),(4,13,10),(5,20,22),(6,17,23),(7,18,24),(8,19,21)])

G:=TransitiveGroup(24,413);

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 4A 4B 4C 4D 6A ··· 6F 6G ··· 6N 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 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 8 8

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 6 type + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 A4 C2×A4 C2×A4 C2×A4 D4×A4 kernel C2×D4×A4 C2×C4×A4 D4×A4 C23×A4 D4×C23 C23×C4 C22×D4 C25 C2×A4 C23 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 4 2 2 2 8 4 2 4 1 1 4 2 2

Matrix representation of C2×D4×A4 in GL7(𝔽13)

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

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0] >;

C2×D4×A4 in GAP, Magma, Sage, TeX

C_2\times D_4\times A_4
% in TeX

G:=Group("C2xD4xA4");
// GroupNames label

G:=SmallGroup(192,1497);
// by ID

G=gap.SmallGroup(192,1497);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,303,530,909]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=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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