Copied to
clipboard

G = C2×A4×D7order 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), 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

C1C2×C14 — C2×A4×D7
C1C7C2×C14C7×A4A4×D7 — C2×A4×D7
C2×C14 — C2×A4×D7
C1C2

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 2A2B2C2D2E2F2G3A3B6A6B6C6D6E6F7A7B7C14A14B14C14D···14I21A···21F42A···42F
order122222223366666677714141414···1421···2142···42
size11337721214444282828282222226···68···88···8

40 irreducible representations

dim111111222233366
type++++++++++
imageC1C2C2C3C6C6D7D14C3×D7C6×D7A4C2×A4C2×A4A4×D7C2×A4×D7
kernelC2×A4×D7A4×D7A4×C14C23×D7C22×D7C22×C14C2×A4A4C23C22D14D7C14C2C1
# reps121242336612133

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

420000
042000
00100
00010
00001
,
10000
01000
00100
000420
000042
,
10000
01000
004200
00010
000042
,
10000
01000
00001
004200
000420
,
2021000
4242000
00100
00010
00001
,
4222000
01000
00100
00010
00001

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

׿
×
𝔽