C2xA4xD7 - GroupNames

G = C₂×A₄×D₇ order 336 = 2⁴·3·7

Direct product of C₂, A₄ and D₇

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

Aliases: C₂×A₄×D₇, C₂³⋊(C₃×D₇), C₂²⋊(C₆×D₇), C₁₄⋊₃(C₂×A₄), C₇⋊₃(C₂²×A₄), (A₄×C₁₄)⋊₂C₂, (C₂³×D₇)⋊₁C₃, (C₇×A₄)⋊₃C₂², (C₂²×C₁₄)⋊₁C₆, (C₂²×D₇)⋊₄C₆, (C₂×C₁₄)⋊₂(C₂×C₆), SmallGroup(336,217)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — C₂×A₄×D₇

Chief series

C₁ — C₇ — C₂×C₁₄ — C₇×A₄ — A₄×D₇ — C₂×A₄×D₇

Lower central

C₂×C₁₄ — C₂×A₄×D₇

Upper central

C₁ — C₂

Generators and relations for C₂×A₄×D₇
G = < a,b,c,d,e,f | a²=b²=c²=d³=e⁷=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd^-1=bc=cb, be=eb, bf=fb, dcd^-1=b, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 580 in 78 conjugacy classes, 21 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₆, C₇, C₂³, C₂³, A₄, C₂×C₆, D₇, D₇, C₁₄, C₁₄, C₂⁴, C₂₁, C₂×A₄, C₂×A₄, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₃×D₇, C₄₂, C₂²×A₄, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₇×A₄, C₆×D₇, C₂³×D₇, A₄×D₇, A₄×C₁₄, C₂×A₄×D₇
Quotients: C₁, C₂, C₃, C₂², C₆, A₄, C₂×C₆, D₇, C₂×A₄, D₁₄, C₃×D₇, C₂²×A₄, C₆×D₇, A₄×D₇, C₂×A₄×D₇

Smallest permutation representation of C₂×A₄×D₇
►On 42 points

Generators in S₄₂

(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)

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

(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)

(1 34 20)(2 35 21)(3 29 15)(4 30 16)(5 31 17)(6 32 18)(7 33 19)(8 36 22)(9 37 23)(10 38 24)(11 39 25)(12 40 26)(13 41 27)(14 42 28)

(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 37 38 39 40 41 42)

(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)

G:=sub<Sym(42)| (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,34,20)(2,35,21)(3,29,15)(4,30,16)(5,31,17)(6,32,18)(7,33,19)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28), (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)(36,37,38,39,40,41,42), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)>;

G:=Group( (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,34,20)(2,35,21)(3,29,15)(4,30,16)(5,31,17)(6,32,18)(7,33,19)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28), (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)(36,37,38,39,40,41,42), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39) );

G=PermutationGroup([[(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(1,34,20),(2,35,21),(3,29,15),(4,30,16),(5,31,17),(6,32,18),(7,33,19),(8,36,22),(9,37,23),(10,38,24),(11,39,25),(12,40,26),(13,41,27),(14,42,28)], [(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),(36,37,38,39,40,41,42)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39)]])

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	6A	6B	6C	6D	6E	6F	7A	7B	7C	14A	14B	14C	14D	···	14I	21A	···	21F	42A	···	42F
order	1	2	2	2	2	2	2	2	3	3	6	6	6	6	6	6	7	7	7	14	14	14	14	···	14	21	···	21	42	···	42
size	1	1	3	3	7	7	21	21	4	4	4	4	28	28	28	28	2	2	2	2	2	2	6	···	6	8	···	8	8	···	8

40 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6
type	+	+	+				+	+			+	+	+	+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	D₇	D₁₄	C₃×D₇	C₆×D₇	A₄	C₂×A₄	C₂×A₄	A₄×D₇	C₂×A₄×D₇
kernel	C₂×A₄×D₇	A₄×D₇	A₄×C₁₄	C₂³×D₇	C₂²×D₇	C₂²×C₁₄	C₂×A₄	A₄	C₂³	C₂²	D₁₄	D₇	C₁₄	C₂	C₁
# reps	1	2	1	2	4	2	3	3	6	6	1	2	1	3	3

Matrix representation of C₂×A₄×D₇ ►in GL₅(𝔽₄₃)

42	0	0	0	0
0	42	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	42	0
0	0	0	0	42

1	0	0	0	0
0	1	0	0	0
0	0	42	0	0
0	0	0	1	0
0	0	0	0	42

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	42	0	0
0	0	0	42	0

20	21	0	0	0
42	42	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

42	22	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(43))| [42,0,0,0,0,0,42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,42,0,0,0,0,0,42],[1,0,0,0,0,0,1,0,0,0,0,0,42,0,0,0,0,0,1,0,0,0,0,0,42],[1,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,42,0,0,1,0,0],[20,42,0,0,0,21,42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[42,0,0,0,0,22,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

C₂×A₄×D₇ in GAP, Magma, Sage, TeX

C_2\times A_4\times D_7

% in TeX

G:=Group("C2xA4xD7");

// GroupNames label

G:=SmallGroup(336,217);

// by ID

G=gap.SmallGroup(336,217);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-7,231,106,10373]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^7=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C2×A4×D7 order 336 = 24·3·7

Direct product of C2, A4 and D7

G = C₂×A₄×D₇ order 336 = 2⁴·3·7

Direct product of C₂, A₄ and D₇