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## G = C24.6(C2×C4)  order 128 = 27

### 6th non-split extension by C24 of C2×C4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.6(C2×C4)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C22.11C24 — C24.6(C2×C4)
 Lower central C1 — C2 — C23 — C24.6(C2×C4)
 Upper central C1 — C2 — C22×C4 — C24.6(C2×C4)
 Jennings C1 — C2 — C2 — C22×C4 — C24.6(C2×C4)

Generators and relations for C24.6(C2×C4)
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=b, f4=d, ab=ba, ac=ca, ad=da, ae=ea, faf-1=abd, bc=cb, fbf-1=bd=db, be=eb, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=cde >

Subgroups: 316 in 136 conjugacy classes, 50 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C4.D4, C2×C22⋊C4, C42⋊C2, C4×D4, C2×M4(2), C22×D4, M4(2)⋊4C4, C2×C4.D4, C22.11C24, C24.6(C2×C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, C24.6(C2×C4)

Character table of C24.6(C2×C4)

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ9 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 i i i -i -i -i -i i linear of order 4 ρ10 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 1 -1 i -i -i -i -i i i i linear of order 4 ρ11 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 -i -i i -i i i -i i linear of order 4 ρ12 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 -i i -i -i i -i i i linear of order 4 ρ13 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 -i -i -i i i i i -i linear of order 4 ρ14 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 1 -1 -i i i i i -i -i -i linear of order 4 ρ15 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 -1 1 i i -i i -i -i i -i linear of order 4 ρ16 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 i -i i i -i i -i -i linear of order 4 ρ17 2 2 -2 -2 2 -2 -2 2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 2 -2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 2 -2 2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 2 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 -2 0 0 0 0 2 2 -2 -2 0 2i 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 2 -2 0 0 0 0 -2 2 -2 2 -2i 0 -2i 2i 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 2 -2 2 -2 0 0 0 0 2 -2 2 -2 2i 0 -2i -2i 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 2 2 -2 -2 0 0 0 0 2 2 -2 -2 0 -2i 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ25 2 2 -2 2 -2 0 0 0 0 2 -2 2 -2 -2i 0 2i 2i 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 -2 -2 0 0 0 0 -2 -2 2 2 0 -2i 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 -2 2 -2 0 0 0 0 -2 2 -2 2 2i 0 2i -2i 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 -2 -2 0 0 0 0 -2 -2 2 2 0 2i 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C24.6(C2×C4)
On 16 points - transitive group 16T233
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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 7 5 3)(2 10)(4 12)(6 14)(8 16)(9 11 13 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,233);

On 16 points - transitive group 16T272
Generators in S16
(1 7)(2 4)(3 5)(6 8)(9 11)(10 16)(12 14)(13 15)
(2 6)(4 8)(10 14)(12 16)
(9 13)(10 14)(11 15)(12 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 11)(2 16 6 12)(3 13)(4 10 8 14)(5 15)(7 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,272);

Matrix representation of C24.6(C2×C4) in GL8(ℤ)

 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
,
 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
,
 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 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0

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,-1,0,0,0,0,0,0,0,0,0,0,0,0,-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],[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],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-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] >;

C24.6(C2×C4) in GAP, Magma, Sage, TeX

C_2^4._6(C_2\times C_4)
% in TeX

G:=Group("C2^4.6(C2xC4)");
// GroupNames label

G:=SmallGroup(128,561);
// by ID

G=gap.SmallGroup(128,561);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2019,718,2028,124]);
// Polycyclic

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

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