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G = C24⋊D4order 128 = 27

1st semidirect product of C24 and D4 acting faithfully

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

Aliases: C241D4, C25.11C22, C24.177C23, C22⋊C45D4, (C22×C4)⋊3D4, C243C49C2, C2.16C2≀C22, C233D42C2, C23.583(C2×D4), C22.40C22≀C2, C23.9D411C2, C23.126(C4○D4), C22.64(C4⋊D4), C22.11(C41D4), C2.26(C232D4), (C22×D4).70C22, (C2×C23⋊C4)⋊8C2, (C2×C22≀C2)⋊1C2, (C2×C22⋊C4).102C22, SmallGroup(128,753)

Series: Derived Chief Lower central Upper central Jennings

C1C24 — C24⋊D4
C1C2C22C23C24C22×D4C2×C22≀C2 — C24⋊D4
C1C2C24 — C24⋊D4
C1C22C24 — C24⋊D4
C1C2C24 — C24⋊D4

Generators and relations for C24⋊D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, faf=ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 896 in 319 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×9], C22 [×3], C22 [×4], C22 [×58], C2×C4 [×21], D4 [×26], C23 [×3], C23 [×4], C23 [×56], C22⋊C4 [×4], C22⋊C4 [×22], C4⋊C4 [×2], C22×C4 [×4], C22×C4 [×3], C2×D4 [×30], C24, C24 [×4], C24 [×10], C23⋊C4 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C22≀C2 [×10], C4⋊D4 [×2], C22.D4 [×2], C22×D4 [×2], C22×D4 [×2], C25, C23.9D4, C243C4, C2×C23⋊C4 [×2], C2×C22≀C2 [×2], C233D4, C24⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, C2≀C22 [×2], C24⋊D4

Character table of C24⋊D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E4F4G4H4I4J
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-11-1111-1-11-11-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-1-11111-11-1-11    linear of order 2
ρ411111111111111-11-111-1-1-1-1-11-1    linear of order 2
ρ51111111111-1-1-1-1111-1-11-1-1-1-111    linear of order 2
ρ6111111111111111-11-1-1-11-11-1-1-1    linear of order 2
ρ711111111111111-1-1-1-1-11-11-11-11    linear of order 2
ρ81111111111-1-1-1-1-11-1-1-1-111111-1    linear of order 2
ρ92222-22-22-2-2000000000200000-2    orthogonal lifted from D4
ρ102222-22-22-2-2000000000-2000002    orthogonal lifted from D4
ρ112-2-222-222-2-200000002-20000000    orthogonal lifted from D4
ρ122222-2-22-22-200000200000000-20    orthogonal lifted from D4
ρ132-2-22-222-2-2222-2-2000000000000    orthogonal lifted from D4
ρ1422222-2-2-2-22000020-2000000000    orthogonal lifted from D4
ρ1522222-2-2-2-220000-202000000000    orthogonal lifted from D4
ρ162-2-222-222-2-20000000-220000000    orthogonal lifted from D4
ρ172222-2-22-22-200000-20000000020    orthogonal lifted from D4
ρ182-2-22-2-2-22220000000000-202000    orthogonal lifted from D4
ρ192-2-22-222-2-22-2-222000000000000    orthogonal lifted from D4
ρ202-2-22-2-2-2222000000000020-2000    orthogonal lifted from D4
ρ212-2-2222-2-22-2000000000002i0-2i00    complex lifted from C4○D4
ρ222-2-2222-2-22-200000000000-2i02i00    complex lifted from C4○D4
ρ234-44-4000000-222-2000000000000    orthogonal lifted from C2≀C22
ρ2444-4-4000000-22-22000000000000    orthogonal lifted from C2≀C22
ρ254-44-40000002-2-22000000000000    orthogonal lifted from C2≀C22
ρ2644-4-40000002-22-2000000000000    orthogonal lifted from C2≀C22

Permutation representations of C24⋊D4
On 16 points - transitive group 16T350
Generators in S16
(1 6)(2 7)(3 14)(4 15)(5 9)(8 12)(10 16)(11 13)
(1 3)(2 9)(4 11)(5 7)(6 14)(8 16)(10 12)(13 15)
(1 3)(2 4)(5 13)(6 14)(7 15)(8 16)(9 11)(10 12)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 15)(6 14)(7 13)(8 16)(9 11)

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

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

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

G:=TransitiveGroup(16,350);

On 16 points - transitive group 16T364
Generators in S16
(1 11)(2 15)(3 12)(4 14)(5 13)(6 9)(7 16)(8 10)
(1 6)(4 7)(9 11)(14 16)
(1 7)(2 8)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(9 11)(14 16)

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

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

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

G:=TransitiveGroup(16,364);

On 16 points - transitive group 16T373
Generators in S16
(1 12)(2 11)(3 13)(4 14)(5 9)(6 10)(7 16)(8 15)
(1 4)(2 8)(3 5)(6 7)(9 13)(10 16)(11 15)(12 14)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 6)(2 5)(3 8)(4 7)(9 15)(10 14)(11 13)(12 16)

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

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

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

G:=TransitiveGroup(16,373);

On 16 points - transitive group 16T392
Generators in S16
(1 9)(2 11)(3 14)(4 16)(5 13)(6 15)(7 12)(8 10)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 16)(11 15)(12 14)
(1 3)(2 4)(5 7)(6 8)(9 14)(10 15)(11 16)(12 13)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 16)(14 15)

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

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

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

G:=TransitiveGroup(16,392);

Matrix representation of C24⋊D4 in GL6(𝔽5)

020000
300000
000010
000001
001000
000100
,
400000
040000
004300
000100
000043
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
010000
400000
004300
001100
000010
000044
,
100000
040000
004300
000100
000010
000001

G:=sub<GL(6,GF(5))| [0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,3,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,1,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C24⋊D4 in GAP, Magma, Sage, TeX

C_2^4\rtimes D_4
% in TeX

G:=Group("C2^4:D4");
// GroupNames label

G:=SmallGroup(128,753);
// by ID

G=gap.SmallGroup(128,753);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,521,4037]);
// Polycyclic

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

Export

Character table of C24⋊D4 in TeX

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