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## G = C2×A4×D7order 336 = 24·3·7

### Direct product of C2, A4 and D7

Aliases: C2×A4×D7, C23⋊(C3×D7), C22⋊(C6×D7), C143(C2×A4), C73(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

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