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G = C23.D4order 64 = 26

2nd non-split extension by C23 of D4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.2D4, (C2×C4).2D4, C22⋊C42C4, (C22×C4)⋊2C4, C4.D4.C2, C23⋊C4.1C2, C23.2(C2×C4), C2.7(C23⋊C4), (C2×D4).2C22, C22.D4.1C2, C22.10(C22⋊C4), SmallGroup(64,33)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.D4
C1C2C22C23C2×D4C22.D4 — C23.D4
C1C2C22C23 — C23.D4
C1C2C22C2×D4 — C23.D4
C1C2C22C2×D4 — C23.D4

Generators and relations for C23.D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad3 >

2C2
4C2
4C2
2C22
2C22
2C4
4C22
4C4
4C22
4C4
8C4
2C2×C4
2C2×C4
4D4
4C8
4C2×C4
4C2×C4
2M4(2)
2C22⋊C4
2C22⋊C4
2C4⋊C4

Character table of C23.D4

 class 12A2B2C2D4A4B4C4D4E4F8A8B
 size 1124444488888
ρ11111111111111    trivial
ρ211111-11-1-1-1-111    linear of order 2
ρ311111111-1-11-1-1    linear of order 2
ρ411111-11-111-1-1-1    linear of order 2
ρ5111-11-1-1-1-ii1i-i    linear of order 4
ρ6111-111-11i-i-1i-i    linear of order 4
ρ7111-11-1-1-1i-i1-ii    linear of order 4
ρ8111-111-11-ii-1-ii    linear of order 4
ρ9222-2-202000000    orthogonal lifted from D4
ρ102222-20-2000000    orthogonal lifted from D4
ρ1144-40000000000    orthogonal lifted from C23⋊C4
ρ124-40002i0-2i00000    complex faithful
ρ134-4000-2i02i00000    complex faithful

Permutation representations of C23.D4
On 16 points - transitive group 16T140
Generators in S16
(1 3)(2 8)(4 6)(5 7)(9 11)(10 12)(13 15)(14 16)
(2 6)(4 8)(9 13)(11 15)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11 3 9)(2 16 8 14)(4 10 6 12)(5 15 7 13)

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

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

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

G:=TransitiveGroup(16,140);

On 16 points - transitive group 16T148
Generators in S16
(2 10)(3 7)(4 16)(6 14)(8 12)(11 15)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 16 10 4)(3 15 7 11)(6 12 14 8)(9 13)

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

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

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

G:=TransitiveGroup(16,148);

On 16 points - transitive group 16T153
Generators in S16
(1 9)(2 10)(3 15)(4 16)(5 13)(6 14)(7 11)(8 12)
(1 5)(3 7)(9 13)(11 15)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 4 9 16)(2 11 10 7)(3 6 15 14)(5 8 13 12)

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

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

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

G:=TransitiveGroup(16,153);

On 16 points - transitive group 16T160
Generators in S16
(2 6)(3 7)(10 14)(11 15)
(2 6)(4 8)(10 14)(12 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 15 6 11)(3 14 7 10)(4 9)(5 12)(8 13)

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

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

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

G:=TransitiveGroup(16,160);

C23.D4 is a maximal subgroup of
C424D4  C426D4  C42.14D4  C22⋊C4⋊F5  (C22×C4)⋊F5
 (C2×D4).D2p: C4○C2≀C4  C24.36D4  C23.(C2×D4)  C42.13D4  (C2×D4).D6  C23.4D12  (C22×C12)⋊C4  (C2×C20).D4 ...
C23.D4 is a maximal quotient of
C23.15M4(2)  C23.2M4(2)  C24.4D4  (C2×C4).Q16  C22⋊C4⋊F5  (C22×C4)⋊F5
 C23.D4p: C23.4D8  C23.4D12  C23.4D20  C23.4D28 ...
 (C2×C4).D4p: (C2×C4).D8  (C2×D4).D6  (C2×C20).D4  (C2×C28).D4 ...
 (C2×D4).D2p: C24.5D4  (C22×C12)⋊C4  (C22×C20)⋊C4  (C22×C28)⋊C4 ...

Matrix representation of C23.D4 in GL4(𝔽5) generated by

0001
0030
0200
1000
,
1000
0400
0040
0001
,
4000
0400
0040
0004
,
0200
1000
0003
0010
,
0400
0004
3000
0020
G:=sub<GL(4,GF(5))| [0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,0],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,2,0,0,0,0,0,0,1,0,0,3,0],[0,0,3,0,4,0,0,0,0,0,0,2,0,4,0,0] >;

C23.D4 in GAP, Magma, Sage, TeX

C_2^3.D_4
% in TeX

G:=Group("C2^3.D4");
// GroupNames label

G:=SmallGroup(64,33);
// by ID

G=gap.SmallGroup(64,33);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,297,255,1444]);
// Polycyclic

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

Export

Subgroup lattice of C23.D4 in TeX
Character table of C23.D4 in TeX

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