D4xC3.A4 - GroupNames

G = D₄×C₃.A₄ order 288 = 2⁵·3²

Direct product of D₄ and C₃.A₄

direct product, metabelian, soluble, monomial

Aliases: D₄×C₃.A₄, C₂⁴⋊₂C₁₈, C₃.(D₄×A₄), C₂²⋊(D₄×C₉), (C₂²×D₄)⋊C₉, (C₂²×C₄)⋊C₁₈, C₁₂.₇(C₂×A₄), (C₃×D₄).₁A₄, (C₂³×C₆).₂C₆, C₂³.₈(C₂×C₁₈), C₆.₁₃(C₂²×A₄), (C₂²×C₁₂).₃C₆, C₄⋊(C₂×C₃.A₄), (D₄×C₂×C₆).C₃, (C₄×C₃.A₄)⋊₃C₂, (C₂×C₆).₁₄(C₂×A₄), (C₂×C₆).₁₂(C₃×D₄), C₂²⋊₂(C₂×C₃.A₄), (C₂²×C₃.A₄)⋊₁C₂, C₂.₂(C₂²×C₃.A₄), (C₂²×C₆).₃₆(C₂×C₆), (C₂×C₃.A₄).₇C₂², SmallGroup(288,344)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — D₄×C₃.A₄

Chief series

C₁ — C₂² — C₂×C₆ — C₂²×C₆ — C₂×C₃.A₄ — C₂²×C₃.A₄ — D₄×C₃.A₄

Lower central

C₂² — C₂³ — D₄×C₃.A₄

Upper central

C₁ — C₆ — C₃×D₄

Generators and relations for D₄×C₃.A₄
G = < a,b,c,d,e,f | a⁴=b²=c³=d²=e²=1, f³=c, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 366 in 116 conjugacy classes, 30 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₂×C₄, D₄, D₄, C₂³, C₂³, C₉, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂⁴, C₁₈, C₂×C₁₂, C₃×D₄, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²×D₄, C₃₆, C₃.A₄, C₂×C₁₈, C₂²×C₁₂, C₆×D₄, C₂³×C₆, D₄×C₉, C₂×C₃.A₄, C₂×C₃.A₄, D₄×C₂×C₆, C₄×C₃.A₄, C₂²×C₃.A₄, D₄×C₃.A₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₉, A₄, C₂×C₆, C₁₈, C₃×D₄, C₂×A₄, C₃.A₄, C₂×C₁₈, C₂²×A₄, D₄×C₉, C₂×C₃.A₄, D₄×A₄, C₂²×C₃.A₄, D₄×C₃.A₄

Smallest permutation representation of D₄×C₃.A₄
►On 36 points

Generators in S₃₆

(1 31 25 16)(2 32 26 17)(3 33 27 18)(4 34 19 10)(5 35 20 11)(6 36 21 12)(7 28 22 13)(8 29 23 14)(9 30 24 15)

(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)

(2 26)(3 27)(5 20)(6 21)(8 23)(9 24)(11 35)(12 36)(14 29)(15 30)(17 32)(18 33)

(1 25)(3 27)(4 19)(6 21)(7 22)(9 24)(10 34)(12 36)(13 28)(15 30)(16 31)(18 33)

(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)

G:=sub<Sym(36)| (1,31,25,16)(2,32,26,17)(3,33,27,18)(4,34,19,10)(5,35,20,11)(6,36,21,12)(7,28,22,13)(8,29,23,14)(9,30,24,15), (10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (2,26)(3,27)(5,20)(6,21)(8,23)(9,24)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,25)(3,27)(4,19)(6,21)(7,22)(9,24)(10,34)(12,36)(13,28)(15,30)(16,31)(18,33), (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)>;

G:=Group( (1,31,25,16)(2,32,26,17)(3,33,27,18)(4,34,19,10)(5,35,20,11)(6,36,21,12)(7,28,22,13)(8,29,23,14)(9,30,24,15), (10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (2,26)(3,27)(5,20)(6,21)(8,23)(9,24)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,25)(3,27)(4,19)(6,21)(7,22)(9,24)(10,34)(12,36)(13,28)(15,30)(16,31)(18,33), (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) );

G=PermutationGroup([[(1,31,25,16),(2,32,26,17),(3,33,27,18),(4,34,19,10),(5,35,20,11),(6,36,21,12),(7,28,22,13),(8,29,23,14),(9,30,24,15)], [(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36)], [(2,26),(3,27),(5,20),(6,21),(8,23),(9,24),(11,35),(12,36),(14,29),(15,30),(17,32),(18,33)], [(1,25),(3,27),(4,19),(6,21),(7,22),(9,24),(10,34),(12,36),(13,28),(15,30),(16,31),(18,33)], [(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)]])

60 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M	6N	9A	···	9F	12A	12B	12C	12D	18A	···	18F	18G	···	18R	36A	···	36F
order	1	2	2	2	2	2	2	2	3	3	4	4	6	6	6	6	6	6	6	6	6	6	6	6	6	6	9	···	9	12	12	12	12	18	···	18	18	···	18	36	···	36
size	1	1	2	2	3	3	6	6	1	1	2	6	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

60 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	3	3	3	3	3	3	6	6
type	+	+	+							+			+	+	+				+
image	C₁	C₂	C₂	C₃	C₆	C₆	C₉	C₁₈	C₁₈	D₄	C₃×D₄	D₄×C₉	A₄	C₂×A₄	C₂×A₄	C₃.A₄	C₂×C₃.A₄	C₂×C₃.A₄	D₄×A₄	D₄×C₃.A₄
kernel	D₄×C₃.A₄	C₄×C₃.A₄	C₂²×C₃.A₄	D₄×C₂×C₆	C₂²×C₁₂	C₂³×C₆	C₂²×D₄	C₂²×C₄	C₂⁴	C₃.A₄	C₂×C₆	C₂²	C₃×D₄	C₁₂	C₂×C₆	D₄	C₄	C₂²	C₃	C₁
# reps	1	1	2	2	2	4	6	6	12	1	2	6	1	1	2	2	2	4	1	2

Matrix representation of D₄×C₃.A₄ ►in GL₅(𝔽₃₇)

0	1	0	0	0
36	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	1	0	0	0
1	0	0	0	0
0	0	36	0	0
0	0	0	36	0
0	0	0	0	36

1	0	0	0	0
0	1	0	0	0
0	0	10	0	0
0	0	0	10	0
0	0	0	0	10

1	0	0	0	0
0	1	0	0	0
0	0	36	36	36
0	0	0	0	1
0	0	0	1	0

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	36	36	36
0	0	1	0	0

26	0	0	0	0
0	26	0	0	0
0	0	0	0	33
0	0	33	0	0
0	0	0	33	0

G:=sub<GL(5,GF(37))| [0,36,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[26,0,0,0,0,0,26,0,0,0,0,0,0,33,0,0,0,0,0,33,0,0,33,0,0] >;

D₄×C₃.A₄ in GAP, Magma, Sage, TeX

D_4\times C_3.A_4

% in TeX

G:=Group("D4xC3.A4");

// GroupNames label

G:=SmallGroup(288,344);

// by ID

G=gap.SmallGroup(288,344);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,142,1531,2666]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=c^3=d^2=e^2=1,f^3=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,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 = D4×C3.A4 order 288 = 25·32

Direct product of D4 and C3.A4

G = D₄×C₃.A₄ order 288 = 2⁵·3²

Direct product of D₄ and C₃.A₄