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G = C24⋊3D4  order 128 = 27

3rd semidirect product of C24 and D4 acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial, rational

Aliases: C24⋊3D4, C42⋊5D4, 2+ 1+4.3C22, C2≀C4⋊3C2, (C2×Q8)⋊4D4, C22⋊C4⋊3D4, C2≀C22⋊2C2, C42⋊3C4⋊6C2, C2.23C2≀C22, D4.9D4⋊2C2, (C2×D4).4C23, C23.16(C2×D4), C23⋊C4.3C22, C24⋊C22⋊2C2, C22≀C2.5C22, C22.47C22≀C2, C4.D4.3C22, C4.4D4.19C22, (C2×C4).16(C2×D4), 2-Sylow(PSL(4,5)), SmallGroup(128,931)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — C2×D4 — C24⋊3D4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C22≀C2 — C24⋊C22 — C24⋊3D4
Lower central
C1 — C2 — C22 — C2×D4 — C24⋊3D4
Upper central
C1 — C2 — C22 — C2×D4 — C24⋊3D4
Jennings
C1 — C2 — C22 — C2×D4 — C24⋊3D4

Generators and relations for C24⋊3D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, faf=ab=ba, ac=ca, ad=da, eae-1=abd, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >

Subgroups: 440 in 135 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×8], C22, C22 [×19], C8, C2×C4, C2×C4 [×9], D4 [×13], Q8 [×3], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×12], M4(2), SD16, Q16, C2×D4, C2×D4 [×9], C2×Q8, C2×Q8, C4○D4 [×2], C24 [×2], C23⋊C4 [×2], C23⋊C4 [×2], C4.D4, C4≀C2, C22≀C2 [×2], C22≀C2 [×4], C4.4D4, C4.4D4 [×4], C8.C22, 2+ 1+4, C2≀C4 [×2], C42⋊3C4, D4.9D4, C2≀C22 [×2], C24⋊C22, C24⋊3D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22, C24⋊3D4

Character table of C24⋊3D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8
 size 11244888488888161616
ρ111111111111111111    trivial
ρ2111111-1111111-1-1-1-1    linear of order 2
ρ311111-1111-11-1-11-11-1    linear of order 2
ρ411111-1-111-11-1-1-11-11    linear of order 2
ρ511111-1-1-111-1-11-111-1    linear of order 2
ρ611111-11-111-1-111-1-11    linear of order 2
ρ71111111-11-1-11-111-1-1    linear of order 2
ρ8111111-1-11-1-11-1-1-111    linear of order 2
ρ92222-200-2-202000000    orthogonal lifted from D4
ρ10222-22200-200-200000    orthogonal lifted from D4
ρ11222-2-20002200-20000    orthogonal lifted from D4
ρ12222-22-200-200200000    orthogonal lifted from D4
ρ13222-2-20002-20020000    orthogonal lifted from D4
ρ142222-2002-20-2000000    orthogonal lifted from D4
ρ1544-40002000000-2000    orthogonal lifted from C2≀C22
ρ1644-4000-20000002000    orthogonal lifted from C2≀C22
ρ178-8000000000000000    orthogonal faithful

Permutation representations of C24⋊3D4
►On 16 points - transitive group 16T342
Generators in S16
(1 6)(2 12)(3 13)(5 10)(8 15)(9 11)

(1 11)(2 12)(3 15)(4 14)(5 10)(6 9)(7 16)(8 13)

(1 6)(4 7)(9 11)(14 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 8)(2 7)(3 6)(4 5)(9 15)(10 14)(11 13)(12 16)
 
G:=sub<Sym(16)| (1,6)(2,12)(3,13)(5,10)(8,15)(9,11), (1,11)(2,12)(3,15)(4,14)(5,10)(6,9)(7,16)(8,13), (1,6)(4,7)(9,11)(14,16), (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), (1,8)(2,7)(3,6)(4,5)(9,15)(10,14)(11,13)(12,16)>;
 
G:=Group( (1,6)(2,12)(3,13)(5,10)(8,15)(9,11), (1,11)(2,12)(3,15)(4,14)(5,10)(6,9)(7,16)(8,13), (1,6)(4,7)(9,11)(14,16), (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), (1,8)(2,7)(3,6)(4,5)(9,15)(10,14)(11,13)(12,16) );
 
G=PermutationGroup([(1,6),(2,12),(3,13),(5,10),(8,15),(9,11)], [(1,11),(2,12),(3,15),(4,14),(5,10),(6,9),(7,16),(8,13)], [(1,6),(4,7),(9,11),(14,16)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,15),(10,14),(11,13),(12,16)])
 
G:=TransitiveGroup(16,342);
 

►On 16 points - transitive group 16T365
Generators in S16
(2 5)(3 13)(4 14)(6 12)(7 9)(11 16)

(2 16)(3 6)(4 9)(5 11)(7 14)(12 13)

(1 10)(2 16)(3 12)(4 14)(5 11)(6 13)(7 9)(8 15)

(1 8)(2 5)(3 6)(4 7)(9 14)(10 15)(11 16)(12 13)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(2 4)(5 7)(9 16)(10 15)(11 14)(12 13)
 
G:=sub<Sym(16)| (2,5)(3,13)(4,14)(6,12)(7,9)(11,16), (2,16)(3,6)(4,9)(5,11)(7,14)(12,13), (1,10)(2,16)(3,12)(4,14)(5,11)(6,13)(7,9)(8,15), (1,8)(2,5)(3,6)(4,7)(9,14)(10,15)(11,16)(12,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,16)(10,15)(11,14)(12,13)>;
 
G:=Group( (2,5)(3,13)(4,14)(6,12)(7,9)(11,16), (2,16)(3,6)(4,9)(5,11)(7,14)(12,13), (1,10)(2,16)(3,12)(4,14)(5,11)(6,13)(7,9)(8,15), (1,8)(2,5)(3,6)(4,7)(9,14)(10,15)(11,16)(12,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,16)(10,15)(11,14)(12,13) );
 
G=PermutationGroup([(2,5),(3,13),(4,14),(6,12),(7,9),(11,16)], [(2,16),(3,6),(4,9),(5,11),(7,14),(12,13)], [(1,10),(2,16),(3,12),(4,14),(5,11),(6,13),(7,9),(8,15)], [(1,8),(2,5),(3,6),(4,7),(9,14),(10,15),(11,16),(12,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(2,4),(5,7),(9,16),(10,15),(11,14),(12,13)])
 
G:=TransitiveGroup(16,365);
 

►On 16 points - transitive group 16T381
Generators in S16
(1 9)(2 14)(3 5)(4 16)(6 8)(7 15)(10 12)(11 13)

(1 5)(2 12)(3 9)(4 8)(6 16)(7 13)(10 14)(11 15)

(2 16)(4 14)(6 12)(8 10)

(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)
 
G:=sub<Sym(16)| (1,9)(2,14)(3,5)(4,16)(6,8)(7,15)(10,12)(11,13), (1,5)(2,12)(3,9)(4,8)(6,16)(7,13)(10,14)(11,15), (2,16)(4,14)(6,12)(8,10), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)>;
 
G:=Group( (1,9)(2,14)(3,5)(4,16)(6,8)(7,15)(10,12)(11,13), (1,5)(2,12)(3,9)(4,8)(6,16)(7,13)(10,14)(11,15), (2,16)(4,14)(6,12)(8,10), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13) );
 
G=PermutationGroup([(1,9),(2,14),(3,5),(4,16),(6,8),(7,15),(10,12),(11,13)], [(1,5),(2,12),(3,9),(4,8),(6,16),(7,13),(10,14),(11,15)], [(2,16),(4,14),(6,12),(8,10)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13)])
 
G:=TransitiveGroup(16,381);
 

►On 16 points - transitive group 16T386
Generators in S16
(1 5)(2 16)(3 7)(6 11)(10 15)(12 13)

(1 5)(2 6)(3 12)(4 9)(7 13)(8 14)(10 15)(11 16)

(1 15)(3 13)(5 10)(7 12)

(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 16)(14 15)
 
G:=sub<Sym(16)| (1,5)(2,16)(3,7)(6,11)(10,15)(12,13), (1,5)(2,6)(3,12)(4,9)(7,13)(8,14)(10,15)(11,16), (1,15)(3,13)(5,10)(7,12), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,16)(14,15)>;
 
G:=Group( (1,5)(2,16)(3,7)(6,11)(10,15)(12,13), (1,5)(2,6)(3,12)(4,9)(7,13)(8,14)(10,15)(11,16), (1,15)(3,13)(5,10)(7,12), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,16)(14,15) );
 
G=PermutationGroup([(1,5),(2,16),(3,7),(6,11),(10,15),(12,13)], [(1,5),(2,6),(3,12),(4,9),(7,13),(8,14),(10,15),(11,16)], [(1,15),(3,13),(5,10),(7,12)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,9),(6,12),(7,11),(8,10),(13,16),(14,15)])
 
G:=TransitiveGroup(16,386);
 

►On 16 points - transitive group 16T394
Generators in S16
(2 15)(3 16)(4 13)(6 11)

(2 15)(3 16)(7 12)(8 9)

(2 15)(4 13)(5 10)(7 12)

(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)
 
G:=sub<Sym(16)| (2,15)(3,16)(4,13)(6,11), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)>;
 
G:=Group( (2,15)(3,16)(4,13)(6,11), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16) );
 
G=PermutationGroup([(2,15),(3,16),(4,13),(6,11)], [(2,15),(3,16),(7,12),(8,9)], [(2,15),(4,13),(5,10),(7,12)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5),(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(12,16)])
 
G:=TransitiveGroup(16,394);
 

Matrix representation of C24⋊3D4 ►in GL8(ℤ)

00100000
000-10000
10000000
0-1000000
00001000
00000-100
00000010
0000000-1
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00001000
00000100
000000-10
0000000-1
01000000
10000000
00010000
00100000
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000

G:=sub<GL(8,Integers())| [0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

C24⋊3D4 in GAP, Magma, Sage, TeX 

C_2^4\rtimes_3D_4
 
% in TeX
 
G:=Group("C2^4:3D4");
 
// GroupNames label
 
G:=SmallGroup(128,931);
 
// by ID
 
G=gap.SmallGroup(128,931);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,723,297,1971,375,4037]);
 
// 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*d,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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