Copied to
clipboard

G = A4×D12order 288 = 25·32

Direct product of A4 and D12

direct product, metabelian, soluble, monomial

Aliases: A4×D12, C4⋊(S3×A4), C31(D4×A4), (C4×A4)⋊3S3, C121(C2×A4), D61(C2×A4), (C3×A4)⋊6D4, (C12×A4)⋊3C2, (C22×D12)⋊C3, (S3×C23)⋊1C6, (C22×C12)⋊1C6, (C2×A4).15D6, C222(C3×D12), C6.3(C22×A4), C23.18(S3×C6), (C6×A4).20C22, (C2×S3×A4)⋊3C2, C2.4(C2×S3×A4), (C2×C6)⋊1(C3×D4), (C22×C4)⋊3(C3×S3), (C22×C6).3(C2×C6), SmallGroup(288,920)

Series: Derived Chief Lower central Upper central

C1C22×C6 — A4×D12
C1C3C2×C6C22×C6C6×A4C2×S3×A4 — A4×D12
C2×C6C22×C6 — A4×D12
C1C2C4

Generators and relations for A4×D12
 G = < a,b,c,d,e | a2=b2=c3=d12=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 778 in 138 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C4, C4, C22, C22 [×14], S3 [×4], C6, C6 [×6], C2×C4 [×2], D4 [×6], C23, C23 [×8], C32, C12, C12 [×3], A4, A4, D6 [×2], D6 [×10], C2×C6, C2×C6 [×4], C22×C4, C2×D4 [×4], C24 [×2], C3×S3 [×2], C3×C6, D12, D12 [×5], C2×C12 [×2], C3×D4, C2×A4, C2×A4 [×3], C22×S3 [×8], C22×C6, C22×D4, C3×C12, C3×A4, S3×C6 [×2], C4×A4, C4×A4, C2×D12 [×4], C22×C12, C22×A4 [×2], S3×C23 [×2], C3×D12, S3×A4 [×2], C6×A4, D4×A4, C22×D12, C12×A4, C2×S3×A4 [×2], A4×D12
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, A4, D6, C2×C6, C3×S3, D12, C3×D4, C2×A4 [×3], S3×C6, C22×A4, C3×D12, S3×A4, D4×A4, C2×S3×A4, A4×D12

Smallest permutation representation of A4×D12
On 36 points
Generators in S36
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 19 25)(2 20 26)(3 21 27)(4 22 28)(5 23 29)(6 24 30)(7 13 31)(8 14 32)(9 15 33)(10 16 34)(11 17 35)(12 18 36)
(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)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 27)(28 36)(29 35)(30 34)(31 33)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K12A12B12C12D12E···12J
order122222223333344666666666661212121212···12
size1133661818244882624466882424242422668···8

36 irreducible representations

dim111111222222223336666
type++++++++++++++
imageC1C2C2C3C6C6S3D4D6C3×S3D12C3×D4S3×C6C3×D12A4C2×A4C2×A4S3×A4D4×A4C2×S3×A4A4×D12
kernelA4×D12C12×A4C2×S3×A4C22×D12C22×C12S3×C23C4×A4C3×A4C2×A4C22×C4A4C2×C6C23C22D12C12D6C4C3C2C1
# reps112224111222241121112

Matrix representation of A4×D12 in GL5(𝔽13)

10000
01000
000121
000120
001120
,
10000
01000
001200
001201
001210
,
30000
03000
00010
00001
00100
,
26000
72000
00100
00010
00001
,
117000
72000
00100
00010
00001

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[2,7,0,0,0,6,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,7,0,0,0,7,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×D12 in GAP, Magma, Sage, TeX

A_4\times D_{12}
% in TeX

G:=Group("A4xD12");
// GroupNames label

G:=SmallGroup(288,920);
// by ID

G=gap.SmallGroup(288,920);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,197,92,648,271,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽