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G = C2×C32⋊C4  order 72 = 23·32

Direct product of C2 and C32⋊C4

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

Aliases: C2×C32⋊C4, (C3×C6)⋊C4, C3⋊S3⋊2C4, C32⋊1(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 >

C1
C2
9C2
9C2
2C3
2C3
9C4
9C4
9C22
2C6
2C6
6S3
6S3
6S3
6S3
C32
9C2×C4
6D6
6D6
C3⋊S3
C3×C6
C3⋊S3
C32⋊C4
C2×C3⋊S3
C32⋊C4
C2×C32⋊C4

Character table of C2×C32⋊C4

 class 12A2B2C3A3B4A4B4C4D6A6B
 size 119944999944
ρ1111111111111    trivial
ρ2111111-1-1-1-111    linear of order 2
ρ31-1-11111-11-1-1-1    linear of order 2
ρ41-1-1111-11-11-1-1    linear of order 2
ρ511-1-111ii-i-i11    linear of order 4
ρ611-1-111-i-iii11    linear of order 4
ρ71-11-111i-i-ii-1-1    linear of order 4
ρ81-11-111-iii-i-1-1    linear of order 4
ρ94400-210000-21    orthogonal lifted from C32⋊C4
ρ104-4001-20000-12    orthogonal faithful
ρ1144001-200001-2    orthogonal lifted from C32⋊C4
ρ124-400-2100002-1    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⋊S3⋊3C8  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

-1000
0-100
00-10
000-1
,
0100
-1-100
0001
00-1-1
,
1000
0100
00-1-1
0010
,
00-10
000-1
1000
-1-100
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
 

Export

Subgroup lattice of C2×C32⋊C4 in TeX
Character table of C2×C32⋊C4 in TeX

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