C2xC7:S4 - GroupNames

G = C₂×C₇⋊S₄ order 336 = 2⁴·3·7

Direct product of C₂ and C₇⋊S₄

direct product, non-abelian, soluble, monomial

Aliases: C₂×C₇⋊S₄, C₁₄⋊S₄, C₂³⋊D₂₁, C₂²⋊D₄₂, A₄⋊₂D₁₄, C₇⋊₂(C₂×S₄), (C₂×A₄)⋊D₇, (C₂×C₁₄)⋊₃D₆, (A₄×C₁₄)⋊₁C₂, (C₇×A₄)⋊₂C₂², (C₂²×C₁₄)⋊₂S₃, SmallGroup(336,215)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₇×A₄ — C₂×C₇⋊S₄

Chief series

C₁ — C₂² — C₂×C₁₄ — C₇×A₄ — C₇⋊S₄ — C₂×C₇⋊S₄

Lower central

C₇×A₄ — C₂×C₇⋊S₄

Upper central

C₁ — C₂

Generators and relations for C₂×C₇⋊S₄
G = < a,b,c,d,e,f | a²=b⁷=c²=d²=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b^-1, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 628 in 66 conjugacy classes, 15 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₇, C₂×C₄, D₄, C₂³, C₂³, A₄, D₆, D₇, C₁₄, C₁₄, C₂×D₄, C₂₁, S₄, C₂×A₄, Dic₇, D₁₄, C₂×C₁₄, C₂×C₁₄, D₂₁, C₄₂, C₂×S₄, C₂×Dic₇, C₇⋊D₄, C₂²×D₇, C₂²×C₁₄, C₇×A₄, D₄₂, C₂×C₇⋊D₄, C₇⋊S₄, A₄×C₁₄, C₂×C₇⋊S₄
Quotients: C₁, C₂, C₂², S₃, D₆, D₇, S₄, D₁₄, D₂₁, C₂×S₄, D₄₂, C₇⋊S₄, C₂×C₇⋊S₄

Smallest permutation representation of C₂×C₇⋊S₄
►On 42 points

Generators in S₄₂

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

(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 17)(2 18)(3 19)(4 20)(5 21)(6 15)(7 16)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 36)

(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 36)(22 35)(23 29)(24 30)(25 31)(26 32)(27 33)(28 34)

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

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

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

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

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

34 conjugacy classes

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

34 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

D₇

D₁₄

D₂₁

D₄₂

S₄

C₂×S₄

C₇⋊S₄

C₂×C₇⋊S₄

kernel

C₂×C₇⋊S₄

C₇⋊S₄

A₄×C₁₄

C₂²×C₁₄

C₂×C₁₄

C₂×A₄

A₄

C₂³

C₂²

C₁₄

C₇

C₂

C₁

# reps

Matrix representation of C₂×C₇⋊S₄ ►in GL₅(𝔽₃₃₇)

336	0	0	0	0
0	336	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

336	336	0	0	0
145	144	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	336	0	0
0	0	1	1	0
0	0	0	0	336

1	0	0	0	0
0	1	0	0	0
0	0	336	0	0
0	0	0	336	0
0	0	336	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	335
0	0	0	0	1
0	0	0	336	336

0	228	0	0	0
34	0	0	0	0
0	0	336	335	0
0	0	0	1	0
0	0	0	336	336

G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[336,145,0,0,0,336,144,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,336,1,0,0,0,0,1,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,336,0,0,0,336,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,335,1,336],[0,34,0,0,0,228,0,0,0,0,0,0,336,0,0,0,0,335,1,336,0,0,0,0,336] >;

C₂×C₇⋊S₄ in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes S_4

% in TeX

G:=Group("C2xC7:S4");

// GroupNames label

G:=SmallGroup(336,215);

// by ID

G=gap.SmallGroup(336,215);

# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,2,146,1731,5044,1276,3029,2285]);

// Polycyclic

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

// generators/relations

G = C2×C7⋊S4 order 336 = 24·3·7

Direct product of C2 and C7⋊S4

G = C₂×C₇⋊S₄ order 336 = 2⁴·3·7

Direct product of C₂ and C₇⋊S₄