Copied to
clipboard

## G = C2×C7⋊S4order 336 = 24·3·7

### Direct product of C2 and C7⋊S4

Aliases: C2×C7⋊S4, C14⋊S4, C23⋊D21, C22⋊D42, A42D14, C72(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 1 1 1 2 2 2 2 2 2 3 3 6 6 type + + + + + + + + + + + + + image C1 C2 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 1 2 1 1 1 3 3 6 6 2 2 3 3

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

׿
×
𝔽