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G = C24.9D4  order 128 = 27

9th non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.9D4, C4⋊C4⋊3D4, (C2×D4)⋊2D4, (C2×Q8)⋊2D4, C23⋊C8⋊5C2, C22⋊D8⋊3C2, C2.9C2≀C22, C23⋊2D4⋊2C2, (C22×C4).13D4, C22⋊SD16⋊27C2, C2.8(D4⋊4D4), C2.9(D4⋊D4), C23.517(C2×D4), C4⋊D4.4C22, (C22×C4).6C23, C22.32C24⋊1C2, C22⋊Q8.4C22, C22.21(C4○D8), C23.31D4⋊5C2, C22.SD16⋊11C2, (C22×D4).7C22, C22.127C22≀C2, C22⋊C8.112C22, C22.19(C8⋊C22), C2.C42.15C22, (C2×C4).195(C2×D4), (C2×C22⋊C4).92C22, SmallGroup(128,332)

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

Derived series
C1 — C22×C4 — C24.9D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C22.32C24 — C24.9D4
Lower central
C1 — C22 — C22×C4 — C24.9D4
Upper central
C1 — C22 — C22×C4 — C24.9D4
Jennings
C1 — C2 — C22 — C22×C4 — C24.9D4

Generators and relations for C24.9D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=c, eae-1=ab=ba, ac=ca, ad=da, faf=abc, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce3 >

Subgroups: 484 in 160 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C24, C2.C42, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C2×D8, C2×SD16, C22×D4, C22×D4, C23⋊C8, C22.SD16, C23.31D4, C23⋊2D4, C22⋊D8, C22⋊SD16, C22.32C24, C24.9D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C4○D8, C8⋊C22, D4⋊D4, D4⋊4D4, C2≀C22, C24.9D4

Character table of C24.9D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112288884444888888888
ρ111111111111111111111111    trivial
ρ2111111-1-1-1-1-111-111-111-11-11    linear of order 2
ρ31111111-1-11-111-1-11-11-11-11-1    linear of order 2
ρ4111111-111-11111-1111-1-1-1-1-1    linear of order 2
ρ5111111-11-1-11111-1-1-1-1-11111    linear of order 2
ρ61111111-111-111-1-1-11-1-1-11-11    linear of order 2
ρ7111111-1-11-1-111-11-11-111-11-1    linear of order 2
ρ811111111-1111111-1-1-11-1-1-1-1    linear of order 2
ρ92222-2-2002002-2000-2000000    orthogonal lifted from D4
ρ102222-2-200-2002-20002000000    orthogonal lifted from D4
ρ112222220200-2-2-2-2000000000    orthogonal lifted from D4
ρ122222-2-200000-220020-200000    orthogonal lifted from D4
ρ132222-2-200000-2200-20200000    orthogonal lifted from D4
ρ142222220-2002-2-22000000000    orthogonal lifted from D4
ρ1522-2-2-220000-2i002i00000√-2√2-√-2-√2    complex lifted from C4○D8
ρ1622-2-2-2200002i00-2i00000-√-2√2√-2-√2    complex lifted from C4○D8
ρ1722-2-2-220000-2i002i00000-√-2-√2√-2√2    complex lifted from C4○D8
ρ1822-2-2-2200002i00-2i00000√-2-√2-√-2√2    complex lifted from C4○D8
ρ194-4-4400000000002000-20000    orthogonal lifted from D4⋊4D4
ρ204-44-400200-20000000000000    orthogonal lifted from C2≀C22
ρ214-44-400-20020000000000000    orthogonal lifted from C2≀C22
ρ2244-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ234-4-440000000000-200020000    orthogonal lifted from D4⋊4D4

Permutation representations of C24.9D4
►On 16 points - transitive group 16T389
Generators in S16
(1 5)(3 13)(4 10)(7 9)(8 14)(11 15)

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

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

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

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

(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)
 
G:=sub<Sym(16)| (1,5)(3,13)(4,10)(7,9)(8,14)(11,15), (1,5)(2,12)(3,7)(4,14)(6,16)(8,10)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)>;
 
G:=Group( (1,5)(3,13)(4,10)(7,9)(8,14)(11,15), (1,5)(2,12)(3,7)(4,14)(6,16)(8,10)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12) );
 
G=PermutationGroup([[(1,5),(3,13),(4,10),(7,9),(8,14),(11,15)], [(1,5),(2,12),(3,7),(4,14),(6,16),(8,10),(9,13),(11,15)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,11),(2,10),(3,9),(4,16),(5,15),(6,14),(7,13),(8,12)]])
 
G:=TransitiveGroup(16,389);
 

Matrix representation of C24.9D4 ►in GL8(ℤ)

-10000000
0-1000000
00100000
00010000
0000-1000
00000100
00000010
00000-1-1-1
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
00000010
00001101
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
00010000
00100000
10000000
0-1000000
00001112
000000-10
0000-1000
00000-10-1
,
10000000
0-1000000
00010000
00100000
0000-1000
00000100
00001112
00000-10-1

G:=sub<GL(8,Integers())| [-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,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-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,-1,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-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],[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],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,1,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,2,0,0,-1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,1,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1] >;
 

C24.9D4 in GAP, Magma, Sage, TeX 

C_2^4._9D_4
 
% in TeX
 
G:=Group("C2^4.9D4");
 
// GroupNames label
 
G:=SmallGroup(128,332);
 
// by ID
 
G=gap.SmallGroup(128,332);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,520,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=c,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f=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=c*e^3>;
 
// generators/relations
 

Export

Character table of C24.9D4 in TeX

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