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G = A4×C32⋊C4order 432 = 24·33

Direct product of A4 and C32⋊C4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4×C32⋊C4, C62⋊C12, C322(C4×A4), (C32×A4)⋊2C4, C22⋊(C3×C32⋊C4), (A4×C3⋊S3).2C2, C3⋊S3.2(C2×A4), (C22×C32⋊C4)⋊C3, (C22×C3⋊S3).2C6, SmallGroup(432,744)

Series: Derived Chief Lower central Upper central

C1C62 — A4×C32⋊C4
C1C32C62C22×C3⋊S3A4×C3⋊S3 — A4×C32⋊C4
C62 — A4×C32⋊C4
C1

Generators and relations for A4×C32⋊C4
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e3=f4=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef-1=de=ed, fdf-1=d-1e >

Subgroups: 652 in 70 conjugacy classes, 12 normal (all characteristic)
C1, C2 [×3], C3 [×5], C4 [×2], C22, C22 [×2], S3 [×4], C6 [×3], C2×C4 [×2], C23, C32, C32 [×4], C12, A4, A4 [×2], D6 [×4], C2×C6 [×2], C22×C4, C3×S3 [×2], C3⋊S3, C3⋊S3, C3×C6, C2×A4, C22×S3 [×2], C33, C32⋊C4, C32⋊C4, C3×A4 [×4], C2×C3⋊S3 [×2], C62, C4×A4, C3×C3⋊S3, S3×A4 [×2], C2×C32⋊C4 [×2], C22×C3⋊S3, C3×C32⋊C4, C32×A4, C22×C32⋊C4, A4×C3⋊S3, A4×C32⋊C4
Quotients: C1, C2, C3, C4, C6, C12, A4, C2×A4, C32⋊C4, C4×A4, C3×C32⋊C4, A4×C32⋊C4

Character table of A4×C32⋊C4

 class 12A2B2C3A3B3C3D3E3F3G3H4A4B4C4D6A6B6C6D12A12B12C12D
 size 139274444161616169927271212363636363636
ρ1111111111111111111111111    trivial
ρ2111111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ31111ζ321ζ31ζ32ζ3ζ3ζ32-1-1-1-111ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ41111ζ31ζ321ζ3ζ32ζ32ζ3111111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ51111ζ31ζ321ζ3ζ32ζ32ζ3-1-1-1-111ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ61111ζ321ζ31ζ32ζ3ζ3ζ32111111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ711-1-111111111-iii-i11-1-1-i-iii    linear of order 4
ρ811-1-111111111i-i-ii11-1-1ii-i-i    linear of order 4
ρ911-1-1ζ321ζ31ζ32ζ3ζ3ζ32i-i-ii11ζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    linear of order 12
ρ1011-1-1ζ31ζ321ζ3ζ32ζ32ζ3i-i-ii11ζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    linear of order 12
ρ1111-1-1ζ321ζ31ζ32ζ3ζ3ζ32-iii-i11ζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    linear of order 12
ρ1211-1-1ζ31ζ321ζ3ζ32ζ32ζ3-iii-i11ζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    linear of order 12
ρ133-13-10303000033-1-1-1-1000000    orthogonal lifted from A4
ρ143-13-103030000-3-311-1-1000000    orthogonal lifted from C2×A4
ρ153-1-31030300003i-3ii-i-1-1000000    complex lifted from C4×A4
ρ163-1-3103030000-3i3i-ii-1-1000000    complex lifted from C4×A4
ρ174400414-2-21-2100001-2000000    orthogonal lifted from C32⋊C4
ρ1844004-2411-21-20000-21000000    orthogonal lifted from C32⋊C4
ρ194400-2-2-3-2-2+2-31ζ321--3ζ31+-30000-21000000    complex lifted from C3×C32⋊C4
ρ204400-2+2-31-2-2-3-21--3ζ321+-3ζ300001-2000000    complex lifted from C3×C32⋊C4
ρ214400-2-2-31-2+2-3-21+-3ζ31--3ζ3200001-2000000    complex lifted from C3×C32⋊C4
ρ224400-2+2-3-2-2-2-31ζ31+-3ζ321--30000-21000000    complex lifted from C3×C32⋊C4
ρ2312-4000-603000000002-1000000    orthogonal faithful
ρ2412-400030-600000000-12000000    orthogonal faithful

Permutation representations of A4×C32⋊C4
On 24 points - transitive group 24T1337
Generators in S24
(1 6)(2 5)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 20)(14 17)(15 18)(16 19)
(1 8)(2 7)(3 5)(4 6)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)
(1 6 4)(2 5 3)(9 22 19)(10 23 20)(11 24 17)(12 21 18)
(1 11 9)(2 10 12)(3 20 18)(4 17 19)(5 23 21)(6 24 22)(7 13 15)(8 14 16)
(2 12 10)(3 18 20)(5 21 23)(7 15 13)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

G:=sub<Sym(24)| (1,6)(2,5)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,20)(14,17)(15,18)(16,19), (1,8)(2,7)(3,5)(4,6)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23), (1,6,4)(2,5,3)(9,22,19)(10,23,20)(11,24,17)(12,21,18), (1,11,9)(2,10,12)(3,20,18)(4,17,19)(5,23,21)(6,24,22)(7,13,15)(8,14,16), (2,12,10)(3,18,20)(5,21,23)(7,15,13), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;

G:=Group( (1,6)(2,5)(3,7)(4,8)(9,22)(10,23)(11,24)(12,21)(13,20)(14,17)(15,18)(16,19), (1,8)(2,7)(3,5)(4,6)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23), (1,6,4)(2,5,3)(9,22,19)(10,23,20)(11,24,17)(12,21,18), (1,11,9)(2,10,12)(3,20,18)(4,17,19)(5,23,21)(6,24,22)(7,13,15)(8,14,16), (2,12,10)(3,18,20)(5,21,23)(7,15,13), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );

G=PermutationGroup([(1,6),(2,5),(3,7),(4,8),(9,22),(10,23),(11,24),(12,21),(13,20),(14,17),(15,18),(16,19)], [(1,8),(2,7),(3,5),(4,6),(9,16),(10,13),(11,14),(12,15),(17,24),(18,21),(19,22),(20,23)], [(1,6,4),(2,5,3),(9,22,19),(10,23,20),(11,24,17),(12,21,18)], [(1,11,9),(2,10,12),(3,20,18),(4,17,19),(5,23,21),(6,24,22),(7,13,15),(8,14,16)], [(2,12,10),(3,18,20),(5,21,23),(7,15,13)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)])

G:=TransitiveGroup(24,1337);

Matrix representation of A4×C32⋊C4 in GL7(𝔽13)

12000000
12010000
12100000
0001000
0000100
0000010
0000001
,
01210000
01200000
11200000
0001000
0000100
0000010
0000001
,
0010000
1000000
0100000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
000121200
0001000
000001212
0000010
,
1000000
0100000
0010000
0001000
0000100
0000001
000001212
,
5000000
0500000
0050000
0000010
0000001
0001000
000121200

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

A4×C32⋊C4 in GAP, Magma, Sage, TeX

A_4\times C_3^2\rtimes C_4
% in TeX

G:=Group("A4xC3^2:C4");
// GroupNames label

G:=SmallGroup(432,744);
// by ID

G=gap.SmallGroup(432,744);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-3,3,42,514,221,10085,691,9414,2372]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^3=f^4=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*e*f^-1=d*e=e*d,f*d*f^-1=d^-1*e>;
// generators/relations

Export

Character table of A4×C32⋊C4 in TeX

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