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G = C23⋊C4order 32 = 25

The semidirect product of C23 and C4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C23⋊C4, C22.2D4, C23.1C22, (C2×C4)⋊C4, C22⋊C41C2, (C2×D4).1C2, C22.2(C2×C4), C2.3(C22⋊C4), SmallGroup(32,6)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23⋊C4
C1C2C22C23C2×D4 — C23⋊C4
C1C2C22 — C23⋊C4
C1C2C23 — C23⋊C4
C1C2C23 — C23⋊C4

Generators and relations for C23⋊C4
 G = < a,b,c,d | a2=b2=c2=d4=1, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >

2C2
2C2
2C2
4C2
2C22
2C4
4C22
4C4
4C22
4C4
2C2×C4
2D4
2C2×C4
2D4

Character table of C23⋊C4

 class 12A2B2C2D2E4A4B4C4D4E
 size 11222444444
ρ111111111111    trivial
ρ211111-11-1-11-1    linear of order 2
ρ3111111-11-1-1-1    linear of order 2
ρ411111-1-1-11-11    linear of order 2
ρ511-1-111i-1-i-ii    linear of order 4
ρ611-1-11-1i1i-i-i    linear of order 4
ρ711-1-111-i-1ii-i    linear of order 4
ρ811-1-11-1-i1-iii    linear of order 4
ρ922-22-2000000    orthogonal lifted from D4
ρ10222-2-2000000    orthogonal lifted from D4
ρ114-4000000000    orthogonal faithful

Permutation representations of C23⋊C4
On 8 points - transitive group 8T19
Generators in S8
(1 2)(3 5)(4 8)(6 7)
(1 3)(2 5)(4 7)(6 8)
(1 6)(2 7)(3 8)(4 5)
(1 2 3 4)(5 6 7 8)

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

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

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

G:=TransitiveGroup(8,19);

On 8 points - transitive group 8T20
Generators in S8
(2 7)(3 8)
(2 7)(4 5)
(1 6)(2 7)(3 8)(4 5)
(1 2 3 4)(5 6 7 8)

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

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

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

G:=TransitiveGroup(8,20);

On 8 points - transitive group 8T21
Generators in S8
(1 5)(2 6)(3 8)(4 7)
(1 4)(5 7)
(1 4)(2 3)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)

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

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

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

G:=TransitiveGroup(8,21);

On 16 points - transitive group 16T33
Generators in S16
(1 10)(2 9)(3 5)(4 8)(6 16)(7 15)(11 14)(12 13)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,33);

On 16 points - transitive group 16T52
Generators in S16
(2 11)(3 13)(4 6)(5 12)(8 16)(9 14)
(1 7)(2 11)(3 5)(4 9)(6 14)(8 16)(10 15)(12 13)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,52);

On 16 points - transitive group 16T53
Generators in S16
(1 10)(2 8)(3 5)(4 9)(6 14)(7 15)(11 16)(12 13)
(2 16)(4 14)(6 9)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,53);

Polynomial with Galois group C23⋊C4 over ℚ
actionf(x)Disc(f)
8T19x8-14x6+36x4-28x2+4232·174
8T20x8-4x7-5x6+24x5+14x4-36x3-20x2+6x+128·56·612
8T21x8-14x6+39x4-32x2+8227·174

Matrix representation of C23⋊C4 in GL4(ℤ) generated by

1000
0-100
0010
000-1
,
1000
0100
00-10
000-1
,
-1000
0-100
00-10
000-1
,
0010
0001
0100
1000
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;

C23⋊C4 in GAP, Magma, Sage, TeX

C_2^3\rtimes C_4
% in TeX

G:=Group("C2^3:C4");
// GroupNames label

G:=SmallGroup(32,6);
// by ID

G=gap.SmallGroup(32,6);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,302,248]);
// Polycyclic

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

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