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## G = C6×C4.D4order 192 = 26·3

### Direct product of C6 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4.D4 — C6×C4.D4
 Lower central C1 — C2 — C22 — C6×C4.D4
 Upper central C1 — C2×C6 — C22×C12 — C6×C4.D4

Generators and relations for C6×C4.D4
G = < a,b,c,d | a6=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 370 in 186 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C4.D4, C2×M4(2), C22×D4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C6×D4, C6×D4, C23×C6, C2×C4.D4, C3×C4.D4, C6×M4(2), D4×C2×C6, C6×C4.D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C4.D4, C2×C22⋊C4, C3×C22⋊C4, C22×C12, C6×D4, C2×C4.D4, C3×C4.D4, C6×C22⋊C4, C6×C4.D4

Smallest permutation representation of C6×C4.D4
On 48 points
Generators in S48
(1 10 39 41 19 27)(2 11 40 42 20 28)(3 12 33 43 21 29)(4 13 34 44 22 30)(5 14 35 45 23 31)(6 15 36 46 24 32)(7 16 37 47 17 25)(8 9 38 48 18 26)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)
(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 48 3 46 5 44 7 42)(2 41 8 43 6 45 4 47)(9 21 15 23 13 17 11 19)(10 18 12 24 14 22 16 20)(25 40 27 38 29 36 31 34)(26 33 32 35 30 37 28 39)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 C3×D4 C4.D4 C3×C4.D4 kernel C6×C4.D4 C3×C4.D4 C6×M4(2) D4×C2×C6 C2×C4.D4 C6×D4 C23×C6 C4.D4 C2×M4(2) C22×D4 C2×D4 C24 C2×C12 C2×C4 C6 C2 # reps 1 4 2 1 2 4 4 8 4 2 8 8 4 8 2 4

Matrix representation of C6×C4.D4 in GL6(𝔽73)

 65 0 0 0 0 0 0 65 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 27 27 0 0 72 0 46 0 0 0 0 0 72 72 0 0 0 0 2 1
,
 72 72 0 0 0 0 2 1 0 0 0 0 0 0 0 46 0 0 0 0 0 27 1 0 0 0 72 1 0 27 0 0 0 71 0 46
,
 1 1 0 0 0 0 0 72 0 0 0 0 0 0 46 0 72 72 0 0 27 0 0 1 0 0 1 72 0 46 0 0 71 0 0 27

G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,65,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,27,46,72,2,0,0,27,0,72,1],[72,2,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,46,27,1,71,0,0,0,1,0,0,0,0,0,0,27,46],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,46,27,1,71,0,0,0,0,72,0,0,0,72,0,0,0,0,0,72,1,46,27] >;

C6×C4.D4 in GAP, Magma, Sage, TeX

C_6\times C_4.D_4
% in TeX

G:=Group("C6xC4.D4");
// GroupNames label

G:=SmallGroup(192,844);
// by ID

G=gap.SmallGroup(192,844);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,4204,3036,124]);
// Polycyclic

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

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