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## G = C2×C32⋊C4order 72 = 23·32

### Direct product of C2 and C32⋊C4

Aliases: C2×C32⋊C4, (C3×C6)⋊C4, C3⋊S32C4, C321(C2×C4), C3⋊S3.3C22, (C2×C3⋊S3).2C2, SmallGroup(72,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4
 Lower central C32 — C2×C32⋊C4
 Upper central C1 — C2

Generators and relations for C2×C32⋊C4
G = < a,b,c,d | a2=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Character table of C2×C32⋊C4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B size 1 1 9 9 4 4 9 9 9 9 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 linear of order 2 ρ4 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 1 i i -i -i 1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 -i -i i i 1 1 linear of order 4 ρ7 1 -1 1 -1 1 1 i -i -i i -1 -1 linear of order 4 ρ8 1 -1 1 -1 1 1 -i i i -i -1 -1 linear of order 4 ρ9 4 4 0 0 -2 1 0 0 0 0 -2 1 orthogonal lifted from C32⋊C4 ρ10 4 -4 0 0 1 -2 0 0 0 0 -1 2 orthogonal faithful ρ11 4 4 0 0 1 -2 0 0 0 0 1 -2 orthogonal lifted from C32⋊C4 ρ12 4 -4 0 0 -2 1 0 0 0 0 2 -1 orthogonal faithful

Permutation representations of C2×C32⋊C4
On 12 points - transitive group 12T40
Generators in S12
(1 4)(2 3)(5 12)(6 9)(7 10)(8 11)
(1 5 7)(2 8 6)(3 11 9)(4 12 10)
(2 6 8)(3 9 11)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)

G:=sub<Sym(12)| (1,4)(2,3)(5,12)(6,9)(7,10)(8,11), (1,5,7)(2,8,6)(3,11,9)(4,12,10), (2,6,8)(3,9,11), (1,2)(3,4)(5,6,7,8)(9,10,11,12)>;

G:=Group( (1,4)(2,3)(5,12)(6,9)(7,10)(8,11), (1,5,7)(2,8,6)(3,11,9)(4,12,10), (2,6,8)(3,9,11), (1,2)(3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([[(1,4),(2,3),(5,12),(6,9),(7,10),(8,11)], [(1,5,7),(2,8,6),(3,11,9),(4,12,10)], [(2,6,8),(3,9,11)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)]])

G:=TransitiveGroup(12,40);

On 12 points - transitive group 12T41
Generators in S12
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)
(2 5 10)(4 12 7)
(1 8 9)(2 5 10)(3 11 6)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)

G:=sub<Sym(12)| (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (2,5,10)(4,12,7), (1,8,9)(2,5,10)(3,11,6)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;

G:=Group( (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (2,5,10)(4,12,7), (1,8,9)(2,5,10)(3,11,6)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([[(1,3),(2,4),(5,12),(6,9),(7,10),(8,11)], [(2,5,10),(4,12,7)], [(1,8,9),(2,5,10),(3,11,6),(4,12,7)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)]])

G:=TransitiveGroup(12,41);

On 18 points - transitive group 18T27
Generators in S18
(1 2)(3 11)(4 12)(5 13)(6 14)(7 16)(8 17)(9 18)(10 15)
(1 10 8)(2 15 17)(3 9 12)(4 11 18)(5 14 7)(6 16 13)
(1 4 6)(2 12 14)(3 7 15)(5 17 9)(8 18 13)(10 11 16)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

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

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

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

G:=TransitiveGroup(18,27);

On 24 points - transitive group 24T76
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)
(1 20 21)(2 22 17)(3 23 18)(4 19 24)(5 10 14)(6 15 11)(7 16 12)(8 9 13)
(2 17 22)(4 24 19)(6 11 15)(8 13 9)
(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)| (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,10,14)(6,15,11)(7,16,12)(8,9,13), (2,17,22)(4,24,19)(6,11,15)(8,13,9), (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( (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,20,21)(2,22,17)(3,23,18)(4,19,24)(5,10,14)(6,15,11)(7,16,12)(8,9,13), (2,17,22)(4,24,19)(6,11,15)(8,13,9), (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([[(1,5),(2,6),(3,7),(4,8),(9,19),(10,20),(11,17),(12,18),(13,24),(14,21),(15,22),(16,23)], [(1,20,21),(2,22,17),(3,23,18),(4,19,24),(5,10,14),(6,15,11),(7,16,12),(8,9,13)], [(2,17,22),(4,24,19),(6,11,15),(8,13,9)], [(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(24,76);

C2×C32⋊C4 is a maximal subgroup of   S32⋊C4  C3⋊S3.Q8  C2.PSU3(𝔽2)  C4⋊(C32⋊C4)  C62⋊C4  C32⋊F5⋊C2
C2×C32⋊C4 is a maximal quotient of   C3⋊S33C8  C32⋊M4(2)  C4⋊(C32⋊C4)  C62.C4  C62⋊C4  C32⋊F5⋊C2

Polynomial with Galois group C2×C32⋊C4 over ℚ
actionf(x)Disc(f)
12T40x12+2x10-36x8+8x6+416x4-1280x2+1600278·38·514·294
12T41x12-3x9-x6+3x3+1318·59

Matrix representation of C2×C32⋊C4 in GL4(ℤ) generated by

 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1
,
 0 1 0 0 -1 -1 0 0 0 0 0 1 0 0 -1 -1
,
 1 0 0 0 0 1 0 0 0 0 -1 -1 0 0 1 0
,
 0 0 -1 0 0 0 0 -1 1 0 0 0 -1 -1 0 0
G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1],[1,0,0,0,0,1,0,0,0,0,-1,1,0,0,-1,0],[0,0,1,-1,0,0,0,-1,-1,0,0,0,0,-1,0,0] >;

C2×C32⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes C_4
% in TeX

G:=Group("C2xC3^2:C4");
// GroupNames label

G:=SmallGroup(72,45);
// by ID

G=gap.SmallGroup(72,45);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,3,20,1123,93,1604,314]);
// Polycyclic

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

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