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G = C24.D4  order 128 = 27

1st non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.1D4, C23.3D8, C23.3SD16, C2.4C2≀C4, C23⋊C8⋊2C2, (C22×D4)⋊1C4, (C22×C4).8D4, C22.13C4≀C2, C2.C42⋊3C4, C23.9D4⋊4C2, C23⋊2D4.1C2, C2.4(C42⋊C4), C22.54(C23⋊C4), C2.8(C22.SD16), C22.17(D4⋊C4), C23.154(C22⋊C4), (C22×C4).1(C2×C4), (C2×C22⋊C4).82C22, SmallGroup(128,75)

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

Derived series
C1 — C22×C4 — C24.D4
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23⋊2D4 — C24.D4
Lower central
C1 — C2 — C23 — C22×C4 — C24.D4
Upper central
C1 — C22 — C23 — C2×C22⋊C4 — C24.D4
Jennings
C1 — C22 — C23 — C2×C22⋊C4 — C24.D4

Generators and relations for C24.D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=dc=cd, f2=a, ab=ba, ac=ca, ad=da, eae-1=abcd, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=acde3 >

Subgroups: 388 in 111 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, D4, C23, C23, C22⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4, C22×D4, C22×D4, C23⋊C8, C23.9D4, C23⋊2D4, C24.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, C23⋊C4, D4⋊C4, C4≀C2, C22.SD16, C2≀C4, C42⋊C4, C24.D4

Character table of C24.D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112244884488888888888
ρ111111111111111111111111    trivial
ρ211111111-1-1111-11111-1-1-1-1-1    linear of order 2
ρ3111111111111-11-1-1-111-1-1-1-1    linear of order 2
ρ411111111-1-111-1-1-1-1-11-11111    linear of order 2
ρ5111111-1-111-1-1i-1-i-ii1-1i-i-ii    linear of order 4
ρ6111111-1-111-1-1-i-1ii-i1-1-iii-i    linear of order 4
ρ7111111-1-1-1-1-1-1i1-i-ii11-iii-i    linear of order 4
ρ8111111-1-1-1-1-1-1-i1ii-i11i-i-ii    linear of order 4
ρ92222222200-2-200000-200000    orthogonal lifted from D4
ρ10222222-2-2002200000-200000    orthogonal lifted from D4
ρ112-22-2-22-2200000000000-√2√2-√2√2    orthogonal lifted from D8
ρ122-22-2-22-2200000000000√2-√2√2-√2    orthogonal lifted from D8
ρ132-22-22-20000-2i2i-1-i01-i-1+i1+i000000    complex lifted from C4≀C2
ρ142-22-22-200002i-2i-1+i01+i-1-i1-i000000    complex lifted from C4≀C2
ρ152-22-2-222-200000000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ162-22-22-20000-2i2i1+i0-1+i1-i-1-i000000    complex lifted from C4≀C2
ρ172-22-22-200002i-2i1-i0-1-i1+i-1+i000000    complex lifted from C4≀C2
ρ182-22-2-222-200000000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ1944-4-4000000000-2000020000    orthogonal lifted from C42⋊C4
ρ204-4-4400002-20000000000000    orthogonal lifted from C2≀C4
ρ214-4-440000-220000000000000    orthogonal lifted from C2≀C4
ρ224444-4-400000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-400000000020000-20000    orthogonal lifted from C42⋊C4

Permutation representations of C24.D4
►On 16 points - transitive group 16T330
Generators in S16
(1 5)(2 13)(3 14)(4 8)(6 9)(7 10)(11 15)(12 16)

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

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

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

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

(1 11 5 15)(2 7 13 10)(3 9 14 6)(4 12 8 16)
 
G:=sub<Sym(16)| (1,5)(2,13)(3,14)(4,8)(6,9)(7,10)(11,15)(12,16), (1,12)(2,6)(3,14)(4,8)(5,16)(7,10)(9,13)(11,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,5,15)(2,7,13,10)(3,9,14,6)(4,12,8,16)>;
 
G:=Group( (1,5)(2,13)(3,14)(4,8)(6,9)(7,10)(11,15)(12,16), (1,12)(2,6)(3,14)(4,8)(5,16)(7,10)(9,13)(11,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,5,15)(2,7,13,10)(3,9,14,6)(4,12,8,16) );
 
G=PermutationGroup([[(1,5),(2,13),(3,14),(4,8),(6,9),(7,10),(11,15),(12,16)], [(1,12),(2,6),(3,14),(4,8),(5,16),(7,10),(9,13),(11,15)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,11,5,15),(2,7,13,10),(3,9,14,6),(4,12,8,16)]])
 
G:=TransitiveGroup(16,330);
 

►On 16 points - transitive group 16T339
Generators in S16
(2 15)(3 16)(6 11)(7 12)

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

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

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

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

(1 8)(2 12 15 7)(3 6 16 11)(4 5)(9 10)(13 14)
 
G:=sub<Sym(16)| (2,15)(3,16)(6,11)(7,12), (1,10)(2,6)(3,12)(4,8)(5,14)(7,16)(9,13)(11,15), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,12,15,7)(3,6,16,11)(4,5)(9,10)(13,14)>;
 
G:=Group( (2,15)(3,16)(6,11)(7,12), (1,10)(2,6)(3,12)(4,8)(5,14)(7,16)(9,13)(11,15), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,12,15,7)(3,6,16,11)(4,5)(9,10)(13,14) );
 
G=PermutationGroup([[(2,15),(3,16),(6,11),(7,12)], [(1,10),(2,6),(3,12),(4,8),(5,14),(7,16),(9,13),(11,15)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,8),(2,12,15,7),(3,6,16,11),(4,5),(9,10),(13,14)]])
 
G:=TransitiveGroup(16,339);
 

Matrix representation of C24.D4 ►in GL6(𝔽17)

100000
010000
001000
0011600
000010
0070016
,
1600000
0160000
0016000
0001600
000010
0010001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
030000
1160000
0000150
001201616
008100
0090100
,
0140000
1100000
0000150
0050161
008000
00916100

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,7,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,10,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,11,0,0,0,0,3,6,0,0,0,0,0,0,0,12,8,9,0,0,0,0,1,0,0,0,15,16,0,10,0,0,0,16,0,0],[0,11,0,0,0,0,14,0,0,0,0,0,0,0,0,5,8,9,0,0,0,0,0,16,0,0,15,16,0,10,0,0,0,1,0,0] >;
 

C24.D4 in GAP, Magma, Sage, TeX 

C_2^4.D_4
 
% in TeX
 
G:=Group("C2^4.D4");
 
// GroupNames label
 
G:=SmallGroup(128,75);
 
// by ID
 
G=gap.SmallGroup(128,75);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,184,794,521,2804]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d*c=c*d,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c*d,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=a*c*d*e^3>;
 
// generators/relations
 

Export

Character table of C24.D4 in TeX

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