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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

Aliases: C24.6(C2×C4), (C2×D4).270D4, C23.65(C22×C4), (C22×C4).27C23, M4(2)⋊4C4⋊13C2, C23.35(C22⋊C4), C42⋊C2.5C22, C22.11C24.2C2, C4.2(C22.D4), (C22×D4).12C22, C22.7(C42⋊C2), C2.17(C23.34D4), (C2×M4(2)).164C22, (C2×C4).235(C2×D4), (C2×C22⋊C4).5C4, C22⋊C4.50(C2×C4), (C2×C4.D4).7C2, (C2×C4).312(C4○D4), C22.37(C2×C22⋊C4), SmallGroup(128,561)

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 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11222444422224444444488888888
ρ111111111111111111111111111111    trivial
ρ211111-1-1-1-111111-111-1-1-111-1-111-1-11    linear of order 2
ρ31111111111111-1-1-1-1-1-1-1-1-11-11-11-11    linear of order 2
ρ411111-1-1-1-11111-11-1-1111-1-1-111-1-111    linear of order 2
ρ511111-1-1-1-111111-111-1-1-11-111-1-111-1    linear of order 2
ρ6111111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ711111-1-1-1-11111-11-1-1111-111-1-111-1-1    linear of order 2
ρ81111111111111-1-1-1-1-1-1-1-11-11-11-11-1    linear of order 2
ρ9111111-11-1-1-1-1-111-11-11-1-1iii-i-i-i-ii    linear of order 4
ρ1011111-11-11-1-1-1-11-1-111-11-1i-i-i-i-iiii    linear of order 4
ρ1111111-11-11-1-1-1-1-111-1-11-11-i-ii-iii-ii    linear of order 4
ρ12111111-11-1-1-1-1-1-1-11-11-111-ii-i-ii-iii    linear of order 4
ρ13111111-11-1-1-1-1-111-11-11-1-1-i-i-iiiii-i    linear of order 4
ρ1411111-11-11-1-1-1-11-1-111-11-1-iiiii-i-i-i    linear of order 4
ρ1511111-11-11-1-1-1-1-111-1-11-11ii-ii-i-ii-i    linear of order 4
ρ16111111-11-1-1-1-1-1-1-11-11-111i-iii-ii-i-i    linear of order 4
ρ1722-2-22-2-222-222-20000000000000000    orthogonal lifted from D4
ρ1822-2-222-2-222-2-220000000000000000    orthogonal lifted from D4
ρ1922-2-22-222-22-2-220000000000000000    orthogonal lifted from D4
ρ2022-2-2222-2-2-222-20000000000000000    orthogonal lifted from D4
ρ21222-2-2000022-2-202i002i-2i-2i000000000    complex lifted from C4○D4
ρ2222-22-20000-22-22-2i0-2i2i0002i00000000    complex lifted from C4○D4
ρ2322-22-200002-22-22i0-2i-2i0002i00000000    complex lifted from C4○D4
ρ24222-2-2000022-2-20-2i00-2i2i2i000000000    complex lifted from C4○D4
ρ2522-22-200002-22-2-2i02i2i000-2i00000000    complex lifted from C4○D4
ρ26222-2-20000-2-2220-2i002i2i-2i000000000    complex lifted from C4○D4
ρ2722-22-20000-22-222i02i-2i000-2i00000000    complex lifted from C4○D4
ρ28222-2-20000-2-22202i00-2i-2i2i000000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    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)

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

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
 

Export

Character table of C24.6(C2×C4) in TeX

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