C2xD4xA4 - GroupNames

G = C₂xD₄xA₄ order 192 = 2⁶·3

Direct product of C₂, D₄ and A₄

direct product, metabelian, soluble, monomial

Aliases: C₂xD₄xA₄, C₂⁵:₂C₆, C₄:(C₂²xA₄), C₂²:(C₆xD₄), (D₄xC₂³):C₃, C₂³:₄(C₂xA₄), (C₂³xC₄):₂C₆, C₂⁴:₃(C₂xC₆), C₂³:₄(C₃xD₄), (C₄xA₄):₄C₂², (C₂³xA₄):₁C₂, (C₂²xD₄):₂C₆, C₂.₂(C₂³xA₄), C₂²:₂(C₂²xA₄), (C₂xA₄).₁₂C₂³, (C₂²xA₄):₂C₂², C₂³.₂₉(C₂²xC₆), (C₂xC₄xA₄):₆C₂, (C₂xC₄):₂(C₂xA₄), (C₂²xC₄):(C₂xC₆), SmallGroup(192,1497)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₂xD₄xA₄

Chief series

C₁ — C₂² — C₂³ — C₂xA₄ — C₂²xA₄ — C₂³xA₄ — C₂xD₄xA₄

Lower central

C₂² — C₂³ — C₂xD₄xA₄

Upper central

C₁ — C₂² — C₂xD₄

Generators and relations for C₂xD₄xA₄
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, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₂xC₄, C₂xC₄, D₄, D₄, C₂³, C₂³, C₂³, C₁₂, A₄, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₂⁴, C₂⁴, C₂xC₁₂, C₃xD₄, C₂xA₄, C₂xA₄, C₂xA₄, C₂²xC₆, C₂³xC₄, C₂²xD₄, C₂²xD₄, C₂⁵, C₄xA₄, C₆xD₄, C₂²xA₄, C₂²xA₄, C₂²xA₄, D₄xC₂³, C₂xC₄xA₄, D₄xA₄, C₂³xA₄, C₂xD₄xA₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, A₄, C₂xC₆, C₂xD₄, C₃xD₄, C₂xA₄, C₂²xC₆, C₆xD₄, C₂²xA₄, D₄xA₄, C₂³xA₄, C₂xD₄xA₄

Permutation representations of C₂xD₄xA₄
►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	C₃xD₄	A₄	C₂xA₄	C₂xA₄	C₂xA₄	D₄xA₄
kernel	C₂xD₄xA₄	C₂xC₄xA₄	D₄xA₄	C₂³xA₄	D₄xC₂³	C₂³xC₄	C₂²xD₄	C₂⁵	C₂xA₄	C₂³	C₂xD₄	C₂xC₄	D₄	C₂³	C₂
# reps	1	1	4	2	2	2	8	4	2	4	1	1	4	2	2

Matrix representation of C₂xD₄xA₄ ►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] >;

C₂xD₄xA₄ 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

G = C2xD4xA4 order 192 = 26·3

Direct product of C2, D4 and A4

G = C₂xD₄xA₄ order 192 = 2⁶·3

Direct product of C₂, D₄ and A₄