C4xC2^2:A4 - GroupNames

G = C₄×C₂²⋊A₄ order 192 = 2⁶·3

Direct product of C₄ and C₂²⋊A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: C₄×C₂²⋊A₄, C₂⁴⋊₆C₁₂, C₂⁵.₄C₆, C₂²⋊(C₄×A₄), (C₂⁴×C₄)⋊₂C₃, (C₂²×C₄)⋊₄A₄, C₂³.₂₁(C₂×A₄), C₂.₁(C₂×C₂²⋊A₄), (C₂×C₂²⋊A₄).₃C₂, SmallGroup(192,1505)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₄×C₂²⋊A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂⁵ — C₂×C₂²⋊A₄ — C₄×C₂²⋊A₄

Lower central

C₂⁴ — C₄×C₂²⋊A₄

Upper central

C₁ — C₄

Generators and relations for C₄×C₂²⋊A₄
G = < a,b,c,d,e,f | a⁴=b²=c²=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf^-1=b, fdf^-1=de=ed, fef^-1=d >

Subgroups: 792 in 262 conjugacy classes, 24 normal (9 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₂×C₄, C₂³, C₂³, C₁₂, A₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂×A₄, C₂³×C₄, C₂⁵, C₄×A₄, C₂²⋊A₄, C₂⁴×C₄, C₂×C₂²⋊A₄, C₄×C₂²⋊A₄
Quotients: C₁, C₂, C₃, C₄, C₆, C₁₂, A₄, C₂×A₄, C₄×A₄, C₂²⋊A₄, C₂×C₂²⋊A₄, C₄×C₂²⋊A₄

Permutation representations of C₄×C₂²⋊A₄
►On 24 points - transitive group 24T385

Generators in S₂₄

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

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

(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)

(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)

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

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

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

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

G:=TransitiveGroup(24,385);

32 conjugacy classes

class	1	2A	2B	···	2K	3A	3B	4A	4B	4C	···	4L	6A	6B	12A	12B	12C	12D
order	1	2	2	···	2	3	3	4	4	4	···	4	6	6	12	12	12	12
size	1	1	3	···	3	16	16	1	1	3	···	3	16	16	16	16	16	16

32 irreducible representations

dim	1	1	1	1	1	1	3	3	3
type	+	+					+	+
image	C₁	C₂	C₃	C₄	C₆	C₁₂	A₄	C₂×A₄	C₄×A₄
kernel	C₄×C₂²⋊A₄	C₂×C₂²⋊A₄	C₂⁴×C₄	C₂²⋊A₄	C₂⁵	C₂⁴	C₂²×C₄	C₂³	C₂²
# reps	1	1	2	2	2	4	5	5	10

Matrix representation of C₄×C₂²⋊A₄ ►in GL₆(𝔽₁₃)

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

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

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

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

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

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C₄×C₂²⋊A₄ in GAP, Magma, Sage, TeX

C_4\times C_2^2\rtimes A_4

% in TeX

G:=Group("C4xC2^2:A4");

// GroupNames label

G:=SmallGroup(192,1505);

// by ID

G=gap.SmallGroup(192,1505);

# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,346,641,2028,3541]);

// Polycyclic

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

// generators/relations

G = C4×C22⋊A4 order 192 = 26·3

Direct product of C4 and C22⋊A4

G = C₄×C₂²⋊A₄ order 192 = 2⁶·3

Direct product of C₄ and C₂²⋊A₄