C6xC4.D4 - GroupNames

G = C₆×C₄.D₄ order 192 = 2⁶·3

Direct product of C₆ and C₄.D₄

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

Aliases: C₆×C₄.D₄, C₂⁴.₃C₁₂, C₄.₄₈(C₆×D₄), (C₂×D₄).₆C₁₂, (C₆×D₄).₁₈C₄, (C₂³×C₆).₂C₄, C₁₂.₄₅₅(C₂×D₄), (C₂×C₁₂).₅₁₅D₄, M₄(2)⋊₈(C₂×C₆), (C₂×M₄(2))⋊₈C₆, C₂³.₄(C₂×C₁₂), (C₂²×D₄).₆C₆, (C₆×M₄(2))⋊₂₆C₂, C₁₂.₇₈(C₂²⋊C₄), (C₂×C₁₂).₆₀₆C₂³, (C₆×D₄).₂₈₄C₂², C₂².₈(C₂²×C₁₂), (C₃×M₄(2))⋊₃₇C₂², (C₂²×C₁₂).₄₀₇C₂², (D₄×C₂×C₆).₁₇C₂, (C₂×C₄).₂₁(C₃×D₄), (C₂×C₄).₁₉(C₂×C₁₂), (C₂×D₄).₄₂(C₂×C₆), C₄.₁₀(C₃×C₂²⋊C₄), C₂.₁₄(C₆×C₂²⋊C₄), (C₂×C₄).₁(C₂²×C₆), (C₂×C₁₂).₁₉₂(C₂×C₄), C₆.₁₀₂(C₂×C₂²⋊C₄), (C₂²×C₄).₃₁(C₂×C₆), (C₂²×C₆).₁₁(C₂×C₄), (C₂×C₆).₁₆₁(C₂²×C₄), C₂².₁₈(C₃×C₂²⋊C₄), (C₂×C₆).₁₃₆(C₂²⋊C₄), SmallGroup(192,844)

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

G = C6×C4.D4 order 192 = 26·3

Direct product of C6 and C4.D4

G = C₆×C₄.D₄ order 192 = 2⁶·3

Direct product of C₆ and C₄.D₄