A4xC4oD4 - GroupNames

G = A₄×C₄○D₄ order 192 = 2⁶·3

Direct product of A₄ and C₄○D₄

direct product, metabelian, soluble, monomial

Aliases: A₄×C₄○D₄, (D₄×A₄)⋊₅C₂, D₄⋊₂(C₂×A₄), (Q₈×A₄)⋊₅C₂, Q₈⋊₄(C₂×A₄), C₂⁴.(C₂×C₆), (C₂³×C₄)⋊₃C₆, (C₂²×D₄)⋊₃C₆, C₂.₆(C₂³×A₄), C₄.₈(C₂²×A₄), (C₂²×Q₈)⋊₇C₆, (C₂×A₄).₁₄C₂³, (C₄×A₄).₂₁C₂², C₂².₇(C₂²×A₄), (C₂²×A₄).₂C₂², C₂³.₃₁(C₂²×C₆), (C₂×C₄×A₄)⋊₇C₂, (C₂×C₄)⋊₃(C₂×A₄), C₂²⋊(C₃×C₄○D₄), (C₂²×C₄○D₄)⋊₂C₃, (C₂²×C₄).₅(C₂×C₆), SmallGroup(192,1501)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — A₄×C₄○D₄

Chief series

C₁ — C₂² — C₂³ — C₂×A₄ — C₂²×A₄ — C₂×C₄×A₄ — A₄×C₄○D₄

Lower central

C₂² — C₂³ — A₄×C₄○D₄

Upper central

C₁ — C₄ — C₄○D₄

Generators and relations for A₄×C₄○D₄
G = < a,b,c,d,e,f | a²=b²=c³=d⁴=f²=1, e²=d², cac^-1=ab=ba, ad=da, ae=ea, af=fa, cbc^-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d²e >

Subgroups: 620 in 215 conjugacy classes, 51 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², C₆, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₂, A₄, C₂×C₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂×A₄, C₂×A₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₄×A₄, C₄×A₄, C₃×C₄○D₄, C₂²×A₄, C₂²×C₄○D₄, C₂×C₄×A₄, D₄×A₄, Q₈×A₄, A₄×C₄○D₄
Quotients: C₁, C₂, C₃, C₂², C₆, C₂³, A₄, C₂×C₆, C₄○D₄, C₂×A₄, C₂²×C₆, C₃×C₄○D₄, C₂²×A₄, C₂³×A₄, A₄×C₄○D₄

Permutation representations of A₄×C₄○D₄
►On 24 points - transitive group 24T296

Generators in S₂₄

(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 13 11)(2 14 12)(3 15 9)(4 16 10)(5 18 21)(6 19 22)(7 20 23)(8 17 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 23 3 21)(2 24 4 22)(5 13 7 15)(6 14 8 16)(9 18 11 20)(10 19 12 17)

(5 7)(6 8)(17 19)(18 20)(21 23)(22 24)

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

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

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

G:=TransitiveGroup(24,296);

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	6A	6B	6C	···	6H	12A	12B	12C	12D	12E	···	12J
order	1	2	2	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	4	4	4	4	6	6	6	···	6	12	12	12	12	12	···	12
size	1	1	2	2	2	3	3	6	6	6	4	4	1	1	2	2	2	3	3	6	6	6	4	4	8	···	8	4	4	4	4	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	C₄○D₄	C₃×C₄○D₄	A₄	C₂×A₄	C₂×A₄	C₂×A₄	A₄×C₄○D₄
kernel	A₄×C₄○D₄	C₂×C₄×A₄	D₄×A₄	Q₈×A₄	C₂²×C₄○D₄	C₂³×C₄	C₂²×D₄	C₂²×Q₈	A₄	C₂²	C₄○D₄	C₂×C₄	D₄	Q₈	C₁
# reps	1	3	3	1	2	6	6	2	2	4	1	3	3	1	2

Matrix representation of A₄×C₄○D₄ ►in GL₅(𝔽₁₃)

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	12	12	12

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	9	0	0
0	0	4	4	4
0	0	0	9	0

8	0	0	0	0
0	8	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

5	9	0	0	0
0	8	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

8	4	0	0	0
7	5	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[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],[1,0,0,0,0,0,1,0,0,0,0,0,9,4,0,0,0,0,4,9,0,0,0,4,0],[8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,9,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,7,0,0,0,4,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A₄×C₄○D₄ in GAP, Magma, Sage, TeX

A_4\times C_4\circ D_4

% in TeX

G:=Group("A4xC4oD4");

// GroupNames label

G:=SmallGroup(192,1501);

// by ID

G=gap.SmallGroup(192,1501);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,176,590,530,909]);

// Polycyclic

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

// generators/relations

G = A4×C4○D4 order 192 = 26·3

Direct product of A4 and C4○D4

G = A₄×C₄○D₄ order 192 = 2⁶·3

Direct product of A₄ and C₄○D₄