C2xD4.A4 - GroupNames

G = C₂×D₄.A₄ order 192 = 2⁶·3

Direct product of C₂ and D₄.A₄

Aliases: C₂×D₄.A₄, 2_-¹⁺⁴⋊₂C₆, SL₂(𝔽₃).₁₀C₂³, D₄.₄(C₂×A₄), (C₂×D₄).₂A₄, C₄.A₄⋊₇C₂², C₂.₈(C₂³×A₄), (C₂²×Q₈)⋊₄C₆, (C₂×2_-¹⁺⁴)⋊C₃, C₂³.₂₈(C₂×A₄), C₄.₁₀(C₂²×A₄), Q₈.₃(C₂²×C₆), C₂².₈(C₂²×A₄), (C₂²×SL₂(𝔽₃))⋊₃C₂, (C₂×SL₂(𝔽₃))⋊₂C₂², C₄○D₄⋊(C₂×C₆), (C₂×Q₈)⋊(C₂×C₆), (C₂×C₄○D₄)⋊₂C₆, (C₂×C₄.A₄)⋊₈C₂, (C₂×C₄).₁₃(C₂×A₄), SmallGroup(192,1503)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — Q₈ — C₂×D₄.A₄

Chief series

C₁ — C₂ — Q₈ — SL₂(𝔽₃) — C₂×SL₂(𝔽₃) — C₂²×SL₂(𝔽₃) — C₂×D₄.A₄

Lower central

Q₈ — 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=f³=1, d²=e²=b², 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, ede^-1=b²d, fdf^-1=b²de, fef^-1=d >

Subgroups: 573 in 204 conjugacy classes, 51 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₂, C₂×C₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, SL₂(𝔽₃), C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, 2_-¹⁺⁴, C₂×SL₂(𝔽₃), C₂×SL₂(𝔽₃), C₄.A₄, C₆×D₄, C₂×2_-¹⁺⁴, C₂²×SL₂(𝔽₃), C₂×C₄.A₄, D₄.A₄, C₂×D₄.A₄
Quotients: C₁, C₂, C₃, C₂², C₆, C₂³, A₄, C₂×C₆, C₂×A₄, C₂²×C₆, C₂²×A₄, D₄.A₄, C₂³×A₄, C₂×D₄.A₄

Smallest permutation representation of C₂×D₄.A₄
►On 32 points

Generators in S₃₂

(1 24)(2 21)(3 22)(4 23)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)

(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)

(2 4)(5 7)(9 11)(14 16)(17 19)(21 23)(25 27)(30 32)

(1 18 3 20)(2 19 4 17)(5 14 7 16)(6 15 8 13)(9 23 11 21)(10 24 12 22)(25 30 27 32)(26 31 28 29)

(1 15 3 13)(2 16 4 14)(5 19 7 17)(6 20 8 18)(9 27 11 25)(10 28 12 26)(21 32 23 30)(22 29 24 31)

(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 32 25)(10 29 26)(11 30 27)(12 31 28)

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3A	3B	4A	4B	4C	···	4H	6A	···	6F	6G	···	6N	12A	12B	12C	12D
order	1	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	6	6	4	4	2	2	6	···	6	4	···	4	8	···	8	8	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3	4	4
type	+	+	+	+					+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	A₄	C₂×A₄	C₂×A₄	C₂×A₄	D₄.A₄	D₄.A₄
kernel	C₂×D₄.A₄	C₂²×SL₂(𝔽₃)	C₂×C₄.A₄	D₄.A₄	C₂×2_-¹⁺⁴	C₂²×Q₈	C₂×C₄○D₄	2_-¹⁺⁴	C₂×D₄	C₂×C₄	D₄	C₂³	C₂	C₂
# reps	1	2	1	4	2	4	2	8	1	1	4	2	2	4

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

12	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
0	0	0	0	0	0	1

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

12	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	12	0
0	0	0	0	0	0	12

12	12	12	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	9	3	0	0
0	0	0	3	4	0	0
0	0	0	0	0	9	3
0	0	0	0	0	3	4

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

1	0	0	0	0	0	0
12	12	12	0	0	0	0
0	1	0	0	0	0	0
0	0	0	1	0	0	0
0	0	0	9	3	0	0
0	0	0	0	0	1	0
0	0	0	0	0	9	3

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,9,3,0,0,0,0,0,3,4,0,0,0,0,0,0,0,9,3,0,0,0,0,0,3,4],[0,1,12,0,0,0,0,1,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0],[1,12,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,3] >;

C₂×D₄.A₄ in GAP, Magma, Sage, TeX

C_2\times D_4.A_4

% in TeX

G:=Group("C2xD4.A4");

// GroupNames label

G:=SmallGroup(192,1503);

// by ID

G=gap.SmallGroup(192,1503);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1059,235,172,404,285,124]);

// Polycyclic

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

// generators/relations

G = C2×D4.A4 order 192 = 26·3

Direct product of C2 and D4.A4

G = C₂×D₄.A₄ order 192 = 2⁶·3

Direct product of C₂ and D₄.A₄